A HISTORY
OF
GREEK MATHEMATICS
VOLUME II
Digitized by the Internet Archive
in 2011 with funding from
University of Ottawa
http://www.archive.org/details/historyofgreekm02heat
A HISTORY
OF
GREEK MATHEMATICS
BY
SIR THOMAS HEATH
K.C.B., K.C.V.O., F.R.S.
SC.D. CAMB. ; HON. D.SC. OXFORD
HONORARY FELLOW (FORMERLY FELLOW) OF TRINITY COLLEGE, CAMBRIDGE
' . . . An independent world,
Created out of pure intelligence.'
Wordsworth.
VOLUME II
FROM ARISTARCHUS TO DIOPHANTUS
OXFORD
AT THE CLARENDON PRESS
1921
*3AN 9 1951
OXFORD UNIVERSITY PRESS
London Edinburgh Glasgow Copenhagen
New York Toronto Melbourne Cape Town
Bombay Calcutta Madras Shanghai
HUMPHREY M1LFORD
Publisher to the University
1623 2
CONTENTS OF VOL II
XII. ARISTARCHUS OF SAMOS pages 1-15
XIII. ARCHIMEDES 16-101
Traditions
(a) Astronomy 17-18
(^) Mechanics ........ 18
Summary of main achievements ..... 19-20
Character of treatises 20-22
List of works still extant 22-23
Traces of lost works 23-25
The text of Archimedes 25-27
Contents of The Method 27-34
On the Sphere and Cylinder, I, II 34-50
Cubic equation arising out of II. 4 .... 43-46
(i) Archimedes's own solution 45-46
(ii) Dionysodorus's solution ..... 46
(iii) Diocles's solution of original problem . . 47-49
Measurement of a Circle ........ 50-56
On Conoids and Spheroids ...... 56-64
On Spirals . . . . . . . ... 64-75
On Plane Equilibriums, I, 11 ...... 75-81
TJie Sand-reel oner (Psammites or Arena ri us) . . . 81-85
The Quadrature of the Parol ola ..... 85-91
On Floating Bodies, I, II 91-97
The problem of the crown 92-94
Other works
(a) The Cattle-Problem 97-98
(#) On semi-regular polyhedra 98-101
(y) The Liber Assnmptoriim ..... 101-103
(6) Formula for area of triangle .... 103
Eratosthenes 104-109
Measurement of the Earth 106 108
XIV. CONIC SECTIONS. APOLLONIUS OF PERGA . . 3 10-196
A. History of Conics up to Apollonius . . 110-126
Discovery of the conic sections by Menaechmus . 110-111
Menaechmus's probable procedure . . . 111-116
Works by Aristaeus and Euclid . . . 116-117
'Solid loci' and 'solid problems' . . .117-118
Aristaeus's Solid Loci 118-119
Focus-directrix property known to Euclid . . 119
Proof from Pappus 120 121
Propositions included in Euclid's Conics . . 121-122
Conic sections in Archimedes .... 122-126
VI
CONTENTS
PAGES
XIV. CONTINUED.
B. Apollonius of Perga
The text of the Conies
Apollonius's own account of the Conks
Extent of claim to originality
Great generality of treatment
Analysis of the Conies
Book I
Conies obtained in the most general way from
oblique cone
New names, ' parabola ', ' ellipse ', ' hyperbola '
Fundamental properties equivalent to Cartesian
equations ........
Transition to new diameter and tangent at its
extremity ........
First appearance of principal axes
Book II
Book III
Book IV
BookV
Normals as maxima and minima ....
Number of normals from a point
Propositions leading immediately to determination
of evolute of conic ......
Construction of normals .....
BookVL
Book VII
Other works by Apollonius .
(a) On the Cutting -off of a Ratio (\6yov dnoTOjxr]),
two Books
(3) On the Cutting-off of an Area {\(opiov «7toto/lii/),
two Books .......
(y) On Determinate Section {dia>pi(riJL€vr} rofxr]), two
Books
(8) On Contacts or Tangencies {encKpal), two Books .
(e) Plane Loci, two Books
(£) Neuo-fty {Verging s or Inclinations), two Books .
• {r}) Comparison of dodecahedron wi'h icosahedron
{&) General Treatise ......
(i) On the Cochlias .......
(k) On Unordered Irrationals . . . . .
(X) On the Burning-mirror .
{(i) 'Qkvtokiov ...
Astronomy .........
126-196
126-128
128-133
132-133
133
133-175
133-148
134-138
138-139
139-141
141-147
147-148
148-150
150-157
157-158
158-167
159-163
163-164
164-166
166-167
167-168
168-174
175-194
175-179
179-180
180-181
181-185
185-189
189-192
192
192-193
193
193
194
194
195-196
XV. THE SUCCESSORS OF THE GREAT GEOMETERS
197-234
Nicomedes
Diocles
Perseus
Isoperimetric figures.
Hypsicles .
Dionysodorus .
Posidonius
Zenodorus
199
200-203
203-206
206-213
213-218
218-219
219-222
CONTENTS vii
Geminus pages 222-234
Attempt to prove the Parallel-Postulate . . . 227-230
On Meteorologica of Posidonius 231-232
Introduction to the Phaenomena attributed to Geminus 232-234
XVJ. SOME HANDBOOKS 235-244
Cleomedes, De motu circulars 235-238
Nicomachus 238
Theon of Smyrna, Expositio rerum mathematicarum ad
legendum Platonem utilium 238-244
XVII. TRIGONOMETRY: HIPPARCHUS, MENELAUS, PTO-
LEMY 245-297
Theodosius 245-246
Works by Theodosius 246
Contents of the Spha erica 246-252
No actual trigonometry in Theodosius . . . 250-252
The beginnings of trigonometry 252-253
Hipparchus . 253-260
The work of Hipparchus . . ' . . . . 254-256
First systematic use of trigonometry .... 257-259
Table of chords 259-260
Menelaus 260-273
The Spkaerka of Menelaus 261-273
(a) ' Menelaus's theorem ' for the sphere . . 266-268
(ft) Deductions from Menelaus's theorem . . 268-269
(y) Anharmonic property of four great circles
through one point ..... 269-270
(d) Propositions analogous to Eucl. VI. 3 . . 270
Claudius Ptolemy 273-297
The MaOtinaTiKr) <Tvi>T<igis (Arab. Almagest) . . . 273-286
Commentaries ....... 274
Translations and editions ..... 274-275
Summary of contents ...... 275-276
Trigonometry in Ptolemy ...'... 276-286
(a) Lemma for finding sin 18° and sin 36° . . 277-278
(ft) Equivalent of sin 2 6 + cos 2 6= 1 ... 278
(y) 'Ptolemy's theorem', giving the equivalent of
sin (6 - 0) = sin 6 cos cp - cos 6 sin <£ . . .278-280
(8) Equivalent of sin 2 ^ = -|(1 -cos (9). . . 280-281
(e) Equivalent of cos(8 + cp) = cos^cos^ — sin0sin0 281
(() Method of interpolation based on formula
sin a/sin j 8<n/ i 3(i 7 r >a >ft) . . .281-282
(rj) Table of chords 283
(6) Further use of proportional increase . . 283-284
(i) Plane trigonometry in effect used . . . 284
Spherical trigonometry : formulae in solution of
spherical triangles 284-286
The Analemma 286-292
The Planisphaerium 292-293
The Optics 293-295
A mechanical work, Yiep\ poncov ..... 295
Attempt to prove the Parallel-Postulate . . . 295-297
4/1
Vlll
CONTENTS
XVIII. MENSURATION: HERON OF ALEXANDRIA, pages 298-354
•'))
Controversies as to Heron's date .
Character of works ....
List of treatises
Geometry
(a) Commentary on Euclid's Elements
(/3) The Definitions ....
Mensuration
The' Metrica, Geometrica, Stereometrica, Geodaesia,
Mensurae .....
Contents of the Metrica
Book I. Measurement of areas
(a) Area of scalene triangle
Proof of formula A = \/{s(s — a) (s — b (s — <
(3) Method of approximating to the square root
of a non-square number ....
(y) Quadrilaterals
(8) Regular polygons with 3, 4, 5, 6, 7, 8, 9, 10,
11, or 12 sides
(?) The circle
(£) Segment o'f a circle
(rj) Ellipse, parabolic segment, surface of cylinder,
right cone, sphere and segment of sphere .
Book II. Measurement of volumes
(a) Cone, cylinder, parallelepiped(prism),pyramid
and frustum ......
(3) Wedge-shaped solid (/3cojuio-ko? or (T<fir)vi<TKo<,) .
(y) Frustum of cone, sphere, and segment thereof
(8) Anchor-ring or tore
(f) The two special solids of Archimedes's'Method'
(£) The five regular solids .
Book III. Divisions of figures . .
Approximation to the cube root of a non-cube
number ......
Quadratic equations solved in Heron .
Indeterminate problems in the Geometrica
The Di optra . .
The Mechanics ......
Aristotle's Wheel
The parallelogram of velocities
Motion on an inclined plane .
On the centre of gravity
The five mechanical powers .
Mechanics in daily life : queries and answers
Problems on the centre of gravity, &c. .
The Catoptrica ......
Heron's proof of equality of angles of incidence and
reflection ........
298-306
307-308
308-310
310-314
314-316
316-344
316-320
320-344
320-331
320-321
321-323
323-326
326
326-329
329
330-331
331
331-335
332
332-334
334
334-335
335
335
336-343
341-342
344
344
345-346
346-352
347-348
348-349
349-350
350-351
351
351-352
352
352-354
353-354
XIX. PAPPUS OF ALEXANDRIA
. 355-439
Date of Pappus 356
Works (commentaries) other than the Collection . . 356-357
CONTENTS ix
The Synagoge or Collection
pages 357-439
(a) Character of the work ; wide range . . . 357-358
(j3) List of authors mentioned ..... 358-360
(y) Translations and editions 360-361
(8) Summary of contents . . . . . . 361-439
Book III. Section (1). On the problem of the two
mean proportionals ...... 361-362
Section (2). The theory of means . . . 363-365
Section (3). The ' Paradoxes ' of Erycinus . .365-368
Section (4). The inscribing of the five regular
solids in a sphere ...... 368-369
Book IV. Section (1). Extension of theorem of
Pythagoras 369-371
Section (2). On circles inscribed in the apftrjXos
('shoemaker's knife') . . _ . . .371-377
Sections (3), (4). Methods of squaring the circle
and trisecting any angle ..... 377-386
(a) The Archimedean spiral 377-379
(#) The conchoid of Nicomecles .... 379
( y ) The Qnadratrix \ 379-382
(b) Digression: a spiral on a sphere . . . 382-385
Trisection (or division in any ratio) of any angle 385-386
Section (5). Solution of the vevais of Archimedes,
On Spirals, Prop. 8, by means of conies . . 386-388
Book V. Preface on the sagacity of Bees . . 389-390
Section (1). Isoperimetry after Zenodorus . . 390-393
Section (2). Comparison of volumes of solids having
their surfaces equal. Case of sphere . . . 393-394
Section (3). Digression on semi-regular solids of
Archimedes ........ 394
Section (4). Propositions on the lines of Archimedes,
On the Sphere and Cylinder ..... 394 395
Section (5). Of regular solids with surfaces equal,
that is greater which has more faces . . . 395-396
Book VI .... 396-399
Problem arising out of Euclid's Optics . . . 397-399
Book VII. On the ' Treasury of Analysis ' . . 399-427
Definition of Analysis and Synthesis . . . 400-401
List of works in the ' Treasury of Analysis' . . 401
Description of the treatises ..... 401-404
Anticipation of Guldin's Theorem . . . 403
Lemmas to the different treatises .... 404-426
(a) Lemmas to the Sectio ration Is and Sectio
spatii of Apollonius 404-405
(13) Lemmas to the Determinate Section of
Apollonius ....... 405-412
(y) Lemmas on the Nt-vtreis of Apollonius . . 412 416
(h) Lemmas on the On Contacts of Apollonius . 416-417
(?) Lemmas to the Plane Loci of Apollonius . 417-419
(() Lemmas to the Porisms of Euclid . . . 419-424
(17) Lemmas to the Comes of Apollonius . .424-425
(6) Lemmas to the Surface Loci of Euclid . . 425-426
(t) An unallocated lemma . . . . • . 426-427
Book VIII. Historical preface 427-429
The object of the Book 429-430
On the centre of gravity ..... 430-433
X
CONTENTS
XIX. CONTINUED.
Book VIII (continued)
The inclined plane .... pages 433-434
Construction of a conic through five points . . 434-437
Given two conjugate diameters of an ellipse, to find
the axes . . . . . . . . . 437-438
Problem of seven hexagons in a circle . . . 438-439
Construction of toothed wheels and indented screws 439
XX. ALGEBRA: DIOPHANTUS OF ALEXANDRIA
440-517
Beginnings learnt from Egypt ...... 440
' Hau '-calculations 440 441
Arithmetical epigrams in the Greek Anthology . . 441-443
Indeterminate equations of first degree .... 443
Indeterminate equations of second degree before Dio-
phantus 443-444
Indeterminate equations in Heronian collections . . 444-447
Numerical solution of quadratic equations . . . 448
Works of Diophantus 448-450
The Arithmetica 449-514
The seven lost Books and their place .... 449-450
Relation of ' Porisms ' to A rithmetica .... 451-452
Commentators from Hypatia downwards . . . 453
Translations and editions 453-455
Notation and definitions 455-461
Sign for unknown (= x) and its origin . . . 456-457
Signs for powers of unknown &c 458-459
The sign (/I\) for minus and its meaning . . . 459-460
The methods of Diophantus 462-479
I. Diophantus's treatment of equations . . . 462-476
(A) Determinate equations
(1) Pure determinate equations . . . 462-463
(2) Mixed quadratic equations .... 463-465
(3) Sinmltaneousequationsinvolving quadratics 465
(4) Cubic equation 465
(B) Indeterminate equations
(a) Indeterminate equations of the second degree 466-473
(1) Single equation 466-468
(2) Double equation 468-473
1. Double equations of first degree . 469 472
2. Double equations of second degree 472-473
(b) Indeterminate equations of degree higher
than second 473-476
(1) Single equations 473-475
(2) Double equations 475-476
II. Method of limits . 476-477
III. Method of approximation to limits . , . . 477-479
Porisms and propositions in the Theory of Numbers . 479-484
(a) Theorems on the composition of numbers as the
sum of two squares 481-483
(fi) On numbers which are the sum of three squares . 483
(y) Composition of numbers as the sum of four squares 483-484
Conspectus of Arithmetica, with typical solutions . . 484-514
The treatise on Polygonal Numbers 514-517
CONTENTS
XI
XXI. COMMENTATORS AND BYZANTINES .
Serenus
(a) On the Section of a Cylinder
(3) On the Section of a Cone .
Theon of Alexandria
Commentary on the Syntaxis
Edition of Euclid's Elements
Edition of the Optics of Euclid
Hypatia .....
Porphyry. Iamblichus .
Proclus .....
Commentary on Euclid, Book I
(a) Sources of the Commentary
(ft) Character of the Commentary
Hypotyposis of Astronomical Hypothcse
Commentary on the Republic
Marinus of Neapolis
Domninus of Larissa
Simplicius
Extracts from Eudemus
Eutocius .
Anthemius of Tralles
On burning-mirrors
The Papyrus of Akhinim
Giodaesin of ' Heron the Younger
Michael Psellus
Georgius Pachymeres
Maximus Planudes .
Extraction of the square root
Two problems
Manuel Moschopoulos
Nicolas Rhabdas
Rule for approximating to square root of a
number
Ioannes Pediasimus .
Barlaam .
Isaac Argyrus .
APPENDIX. On Archimedes's p
of a spiral
INDEX OF GREEK WORDS
ENGLISH INDEX .
pages 518 555
519-526
519-522
522-526
526-528
526-527
527-528
528
528-529
529
529-537
530-535
530-532
532-535
535-536
536-537
537-538
538
538-540
539
540-541
541-543
541-543
543-545
545
545-546
546
546-549
547-549
549
549-550
550-554
on-square
oof of the subtangent-property
553-554
554
554-555
555
556-561
.63-569
570-586
XII
ARISTARCHUS OF SAMOS
Historians of mathematics have, as a rule, given too little
attention to Aristarchus of Samos. The reason is no doubt
that he was an astronomer, and therefore it might be supposed
that his work would have no sufficient interest for the mathe-
matician. The Greeks knew better; they called him Aristar-
chus ' the mathematician ', to distinguish him from the host
of other Aristarchuses ; he is also included by Vitruvius
among the few great men who possessed an equally profound
knowledge of all branches of science, geometry, astronomy,
music, &c.
i Men of this type are rare, men such as were, in times past,
Aristarchus of Samos, Philolaus and Archytas of Tarentum,
Apollonius of Perga, Eratosthenes of Cyrene, Archimedes and
Scopinas of Syracuse, who left to posterity many mechanical
and gnomonic appliances which they invented and explained
on mathematical (lit. ' numerical ') principles.' *
That Aristarchus was a very capable geometer is proved by
his extant work On the sizes and distances of the Sun and
Moon which will be noticed later in this chapter : in the
mechanical line he is credited with the discovery of an im-
proved sun-dial, the so-called crKacprj, which had, not a plane,
but a concave hemispherical surface, with a pointer erected
vertically in the middle throwing shadows and so enabling
the direction and the height of the sun to be read off by means
of lines marked on the surface of the hemisphere. He also
wrote on vision, light and colours. His views on the latter
subjects were no doubt largely influenced by his master, Strato
of Lampsacus ; thus Strato held that colours were emanations
from bodies, material molecules, as it were, which imparted to
the intervening air the same colour as that possessed by the
body, while Aristarchus said that colours are .' shapes or forms
1 Vitruvius, De architecture/,, i. 1. 16.
1523.2 B
2 ARISTARCHUS OF SAMOS
stamping the air with impressions like themselves, as it were ',
that ' colours in darkness have no colouring ', and that ' light
is the colour impinging on a substratum '.
Two facts enable us to fix Aristarchus's date approximately.
In 281/280 B.C. he made an observation of the summer
solstice ; and a book of his, presently to be mentioned, was
published before the date of Archimedes's Psammites or Sand-
reckoner, a work written before 216 B.C. Aristarchus, there-
fore, probably lived circa 310-230 B.C., that is, he was older
than Archimedes by about 25 years.
To Aristarchus belongs the high honour of having been
the first to formulate the Copernican hypothesis, which was
then abandoned again until it was revived by Copernicus
himself. His claim to the title of ' the ancient Copernicus ' is
still, in my opinion, quite unshaken, notwithstanding the in-
genious and elaborate arguments brought forward by Schia-
parelli to prove that it was Heraclides of Pontus who first
conceived the heliocentric idea. Heraclides is (along with one
Ecphantus, a Pythagorean) credited with having been the first
to hold that the earth revolves about its own axis every 24
hours, and he was the first to discover that Mercury and Venus
revolve, like satellites, about the sun. But though this proves
that Heraclides came near, if he did not actually reach, the
hypothesis of Tycho Brahe, according to which the earth was
in the centre and the rest of the system, the sun with the
planets revolving round it, revolved round the earth, it does
not suggest that he moved the earth away from the centre.
The contrary is indeed stated by Aetius, who says that ' Hera-
clides and Ecphantus make the earth move, not in the sense of
translation, but by way of turning on an axle, like a wheel,
from west to east, about its own centre '} None of the
champions of Heraclides have been able to meet this positive
statement. But we have conclusive evidence in favour of the
claim of Aristarchus ; indeed, ancient testimony is unanimous
on the point. Not only does Plutarch tell us that Cleanthes
held that Aristarchus ought to be indicted for the impiety of
' putting the Hearth of the Universe in motion ' 2 ; we have the
best possible testimony in the precise statement of a great
1 Aet. iii. 13. 3, Vors. i 3 , p. 341. 8.
2 Plutarch, De facie in orbe lunae, c. 6, pp. 922 f-923 a^
ARISTARCHUS OF SAMOS 3
contemporary, Archimedes. In the Sand-reckoner Archi-
medes has this passage.
1 You [King Gelon] are aware that " universe " is the name
given by most astronomers to the sphere the centre of which
is the centre of the earth, while its radius is equal to the
straight line between the centre of the sun and the centre of
the earth. This is the common account, as you have heard
from astronomers. But Aristarchus brought out a book con-
sisting of certain hypotheses, wherein it appears, as a conse-
quence of the assumptions made, that the universe is many
times greater than the " universe "just mentioned. His hypo-
theses are that the fixed stars and the sun remain unmoved,
that the earth revolves about the sun in the circumference of a
circle, the sun lying in the middle of the orbit, and that the
sphere of the fixed stars, situated about the same centre as the
sun, is so great that the circle in which he supposes the earth
to revolve bears such a proportion to the distance of the fixed
stars as the centre of the sphere bears to its surface.'
(The last statement is a variation of a traditional phrase, for
which there are many parallels (cf . Aristarchus's Hypothesis 2
' that the earth is in the relation of a point and centre to the
sphere in which the moon moves '), and is a method of saying
that the ' universe ' is infinitely great in relation not merely to
the size of the sun but even to the orbit of the earth in its
revolution about it ; the assumption was necessary to Aris-
tarchus in order that he might not have to take account of
parallax.)
Plutarch, in the passage referred to above, also makes it
clear that Aristarchus followed Heraclides in attributing to
the earth the daily rotation about its axis. The bold hypo-
thesis of Aristarchus found few adherents. Seleucus, of
Seleucia on the Tigris, is the only convinced supporter of it of
whom we hear, and it was speedily abandoned altogether,
mainly owing to the great authority of Hipparchus. Nor'do
we find any trace of the heliocentric hypothesis in Aris- %
tarchus's extant work On the sizes and distances of the
Sun and Moon. This is presumably because that work was
written before the hypothesis was formulated in the book
referred to by Archimedes, The geometry of the treatise
is, however, unaffected by the difference between the hypo-
theses.
B 2
4 ARISTARCHUS OF SAMOS
Archimedes also says that it was Aristarchus who dis-
covered that the apparent angular diameter of the sun is about
l/720th part of the zodiac circle, that is to say, half a degree.
We do not know how he arrived at this pretty accurate figure :
but, as he is credited with the invention of the cr/ca0?7, he may
have used this instrument for the purpose. But here again
the discovery must apparently have been later than the trea-
tise On sizes and distances, for the value of the subtended
angle is there assumed to be 2° (Hypothesis 6). How Aris-
tarchus came to assume a value so excessive is uncertain. As
the mathematics of his treatise is not dependent on the actual
value taken, 2° may have been assumed merely by way of
illustration ; or it may have been a guess at the apparent
diameter made before he had thought of attempting to mea-
sure it. Aristarchus assumed that the angular diameters of
the sun and moon at the centre of the earth are equal.
The method of the treatise depends on the just observation,
which is Aristarchus's third ' hypothesis ', that ' when the moon
appears to us halved, the great circle which divides the dark
and the bright portions of the moon is in the direction of our
eye ' ; the effect of this (since the moon receives its light from
the sun), is that at the time of the dichotomy the centres of
the sun, moon and earth form a triangle right-angled at the
centre of the moon. Two other assumptions were necessary :
first, an estimate of the size of the angle of the latter triangle
at the centre of the earth at the moment of dichotomy : this
Aristarchus assumed (Hypothesis 4) to be 'less than a quad-
rant by one-thirtieth of a quadrant', i. e. 87°, again an inaccu-
rate estimate, the true value being 89° 50' ; secondly, an esti-
mate of the breadth of the earth's shadow where the moon
traverses it : this he assumed to be ' the breadth of two
moons ' (Hypothesis 5).
The inaccuracy of the assumptions does not, however, detract
from the mathematical interest of the succeeding investigation.
Here we find the logical sequence of propositions and the abso-
lute rigour of demonstration characteristic of Greek geometry ;
the only remaining drawback would be the practical difficulty
of determining the exact moment when the moon ' appears to
us halved '. The form and style of the book are thoroughly
classical, as befits the period between Euclid and Archimedes ;
ARISTARCHUS OF SAMOS 5
the Greek is even remarkably attractive. The content from
the mathematical point of view is no less interesting, for we
have here the first specimen extant of pure geometry used
with a trigonometrical object, in which respect it is a sort of
forerunner of Archimedes's Measurement of a Circle. Aristar-
chus does not actually evaluate the trigonometrical ratios
on which the ratios of the sizes and distances to be obtained
depend ; he finds limits between which they lie, and that by
means of certain propositions which he assumes without proof,
and which therefore must have been generally known to
mathematicians of his day. These propositions are the equi-
valents of the statements that,
(1) if oc is what we call the circular measure of an angle
and oc is less than \ it, then the ratio sin oc/oc decreases, and the
ratio tan oc/oc increases, as a increases from to J it ;
(2) if /3 be the circular measure of another angle less than
\ it, and oc > /3, then
sin a oc tan oc
sin ft (3 tan fi
Aristarchus of course deals, not with actual circular measures,
sines and tangents, but with angles (expressed not in degrees
but as fractions of right angles), arcs of circles and their
chords. Particular results obtained by Aristarchus are the
equivalent of the following :
^ > sin 3° > fa [Prop. 7]
■is >sinl°>^, [Prop. 11]
1 > cosl° > §§, [Prop. 12]
1 >cos 2 l° > |f. [Prop. 13]
The book consists of eighteen propositions. Beginning with
six hypotheses to the effect already indicated, Aristarchus
declares that he is now in a position to prove
(1) that the distance of the sun from the earth is greater than
eighteen times, but less than twenty times, the distance of the
moon from the earth ;
(2) that the diameter of the sun has the same ratio as afore-
said to the diameter of the moon ;
6 ARISTARCHUS OF SAMOS
(3) that the diameter of the sun has to the diameter of the
earth a ratio greater than 19:3, but less than 43 : 6.
The propositions containing these results are Props. 7, 9
and 15.
Prop. 1 is preliminary, proving that two equal spheres are
comprehended by one cylinder , and two unequal spheres by
one cone with its vertex in the direction of the lesser sphere,
and the cylinder or cone touches the spheres in circles at
right angles to the line of centres. Prop. 2 proves that, if
a sphere be illuminated by another sphere larger than itself,
the illuminated portion is greater than a hemisphere. Prop. 3
shows that the circle in the moon which divides the dark from
the bright portion is least when the cone comprehending the
sun and the moon has its vertex at our eye. The ' dividing
circle ', as we shall call it for short, which was in Hypothesis 3
spoken of as a great circle, is proved in Prop. 4 to be, not
a great circle, but a small circle not perceptibly different
from a great circle. The proof is typical and is worth giving
along with that of some connected propositions (11 and 12).
B is the centre of the moon, A that of the earth, CD the
diameter of the ' dividing circle in the moon ', EF the parallel
diameter in the moon. BA meets the circular section of the
moon through A and EF in G, and CD in L. GH, GK
are arcs each of which is equal to half the arc CE. By
Hypothesis 6 the angle CAD is ' one-fifteenth of a sign' = 2°,
and the angle BAC = 1°.
Now, says Aristarchus,
1°:45°[> tan 1°: tan 45°]
> BC.CA,
and, a fortiori,
BC.BA or BG:BA
< 1:45;
that is, BG<^BA
therefore, a fortiori,
< A GA
BH<^HA.
ARISTARCHUS OF SAMOS
Now
whence
BH:HA[ = sin HAB : sin HBA]
> lHAB.lHBA,
LEAB< &LHBA,
F D
and (taking the doubles) Z HAK < £ Z HBK.
But Z imST = Z #£(7 = ^o R (where E is a right angle) ;
therefore
Z.HAK<iA*R.
3 9^
But 'a magnitude (arc HK) seen under such an angle is
imperceptible to our eye ' ;
therefore, a fortiori, the arcs CE, DF are severally imper-
ceptible to our eye. Q. E. D.
An easy deduction from the same figure is Prop. 12, which
shows that the ratio of CD, the diameter of the 'dividing
circle ', to EF, the diameter of the moon, is < 1 but > §§ .
We have Z EBC = Z BAG = 1° ;
therefore (arc EC) = ^ (arc EG),
and accordingly (arc CG) : (arc GE) = 89 : 90.
Doubling the arcs, we have
(arc CGD) : (arc EGF) = 89 : 90.
But CD:EF> (arc CGD) : (arc EGF)
[equivalent to sin oc /sin ft > oc/ft; where /.CBD = 2 a,
and 2 /3 = 7r] ;
therefore CD.EF [= cos 1°] > 89 : 90,
while obviously CD : EF < 1.
Prop. 11 finds limits to the ratio EF:BA (the ratio of the
diameter of the moon to the distance of its centre from
the centre of the earth) ; the ratio is < 2 : 45 but > 1 : 30.
8 ARISTARCHUS OF SAMOS
The first part follows from the relation found in Prop. 4,
namely BG : BA < 1:45,
for EF = 2 BC.
The second part requires the use of the circle drawn with
centre A and radius AC. This circle is that on which the
ends of the diameter of the ' dividing circle ' move as the moon
moves in a circle about the earth. If r is the radius AC
of this circle, a chord in it equal to r subtends at the centre
A an angle of %R or 60°; and the arc CD subtends at A
an angle of 2°.
But (arc subtended by CD) : (arc subtended by r)
< CD:r,
or 2:60 < CD:r;
that is, CD:CA > 1:30.
And, by similar triangles,
CL:CA = CB:BA, or CD.CA = 2CB: BA = FE.BA.
Therefore FE: BA > 1 : 30.
The proposition which is of the greatest interest on the
whole is Prop. 7, to the effect that the distance of the sun
from the earth is greater than 18 times, but less than 20
times, the distance of the moon from the earth. This result
represents a great improvement on all previous attempts to
estimate the relative distances. The first speculation on the
subject was that of Anaximander {circa 611-545 B.C.), who
seems to have made the distances of the sun and moon from
the earth to be in the ratio 3 : 2. Eudoxus, according to
Archimedes, made the diameter of the sun 9 times that of
the moon, and Phidias, Archimedes's father, 1 2 times ; and,
assuming that the angular diameters of the two bodies are
equal, the ratio of their distances would be the same.
Aristarchus's proof is shortly as follows. A is the centre of
the sun, B that of the earth, and C that of the moon at the
moment of dichotomy, so that the angle ACB is right. ABEF
is a square, and AE is a quadrant of the sun's circular orbit.
Join BF, and bisect the angle FBE by BG, so that
«
IGBE= \R or 22|°.
ARISTARCHUS OF SAMOS
I. Now, by Hypothesis 4, /.ABC = 87°,
so that Z HBE = Z 5ZIC = 3° ;
therefore
Z.GBE:lHBE=iR: 3 \R
= 15:2,
9
so that GE:HE[ = tan ££# : tan #£#] > Z QBE : Z #£#
> 15 :2. (1)
The ratio which has to be proved > 18:1 is AB:BC or
FE:EH.
Now FG:GE=FB:BE,
whence FG 2 : GE 2 = FB 2 : BE 2 =2:1,
and FG:GE = V2:1
> 7:5
(this is the approximation to V2 mentioned by Plato and
known to the Pythagoreans).
10 ARISTARCHUS OF SAMOS
Therefore ' FE.EG > 12 : 5 or 36 : 15.
Compounding this with (1) above, we have
FE.EH > 36:2 or 18:1.
II. To prove BA < 20 BC.
Let BH meet the circle AE in D, and draw DK parallel
to EB. Circumscribe a circle about the triangle BKD, and
let the chord BL be equal to the radius (r) of the circle.
Now Z BDK = L DBF = ^ R,
so that arc BK = ^ (circumference of circle).
Thus (arc BK) : (arc BL) = A : I >
= 1:10.
And (arc BK ) : (arc BL) < BK:r
[this is equivalent to a//3 < sin a/sin ft, where ot < (3 < J77-],
so that r < 10 BK,
and BD < 20 BK.
But BD.BK = AB:BC;
therefore AB < 20 BC. Q. E. D.
The remaining results obtained in the treatise can be
visualized by means of the three figures annexed, which have
reference to the positions of the sun (centre A), the earth
(centre B) and the moon (centre C) during an eclipse. Fig. 1
shows the middle position of the moon relatively to the earth's
shadow which is bounded by the cone comprehending the sun
and the earth. OAT is the arc with centre B along which
move the extremities of the diameter of the dividing circle in
the moon. Fig. 3 shows the same position of the moon in the
middle of the shadow, but on a larger scale. Fig. 2 shows
the moon at the moment when it has just entered the shadow ;
and, as the breadth of the earth's shadow is that of two moons
(Hypothesis 5), the moon in the position shown touches BN at
A^and BL at L, where L is the middle point of the arc ON.
It should be added that, in Fig. 1, UV \& the diameter of the
circle in which the sun is touched by the double cone with B
as vertex, which comprehends both the sun and the moon,
ARISTARCHUS OF SAMOS
11
while Y, Z are the points in which the perpendicular through
A, the centre of the sun, to BA meets the cone enveloping the
sun and the earth.
Fig. 1.
This being premised, the main results obtained are as
follows :
Prop. 13.
(1) ON : (diam. of moon) < 2 : 1
but > 88:45.
12
ARISTARCHUS OF SAMOS
(2) ON: (diam. of sun) < 1 : 9
but > 22: 225.
(3) ON.YZ > 979:10125.
Prop. 14 (Fig. 3).
Prop. 15.
BC:CS> 675:1.
but
(Diam. of sun) : (diam. of earth) > 19:3
< 43:6.
Prop. 17.
(Diam. of earth) : (diam. of moon) > 108 : 43
but < 60: 19.
It is worth while to show how these results are proved.
Prop. 13.
(1) In Fig. 2 it is clear that
ON < 2 LN and, a fortiori, < 2 LP.
The triangles LON, GLN being similar,
0N:NL = NL:LC;
therefore 0N:NL = NL : \ LP
> 89: 45. (by Prop. 12)
ARISTARCHUS OF SAMOS 13
Hence N : LC = ON 2 : NL 2
> 89 2 :45 2 ;
therefore ON: LP > 7921 : 4050
> 88 : 45, says Aristarchus.
[If Jffjy be developed as a continued fraction, we easily
i , • 1 ! 1 i • t • • o , 88 ~]
obtain 1 H , which is in tact
1+21+2 45 J
(2) ON < 2 (diam. of moon).
But (diam. of moon) < ^g (diam. of sun) ; (Prop. 7)
therefore ON < | (diam. of sun).
Again ON: (diam. of moon) > 88 : 45, from above,
and (diam. of moon) : (diam. of sun) > 1 : 20 ; (Prop. 7)
therefore, ex aequali,
ON: (diam. of sun) > 88 : 900 ■
> 22:225.
(3) Since the same cone comprehends the sun and the moon,
the triangle BUV (Fig. 1) and the triangle BLN (Fig. 2) are
similar, and
LN:LP = UV: (diam. of sun)
= WU:UA
= UA:AS
< UA:AY.
But . LN : LP > 89 : 90 ; (Prop. 12)
therefore, a fortiori, UA : AY > 89 : 90.
And UA:AY=2UA:YZ
= (diam. of sun) : YZ.
But ON: (diam. of sun) > 22 : 225 ; (Prop. 13)
therefore, ex aequali,
ON: YZ > 89 x 22 : 90 x 225
> 979:10125.
14 ARISTARCHUS OF SAMOS
Prop. 14 (Fig. 3).
The arcs OM, ML, LP, PX are all equal ; therefore so are
the chords. BM, BP are tangents to the circle MQP, so that
CM is perpendicular to BM, while BM is perpendicular to OL.
Therefore the triangles LOS, CMR are similar.
Therefore SO :MR = SL: RC.
But SO < 2 MR, since OJS T < 2 MP; (Prop. 13)
therefore SL < 2 RC,
and, a fortiori, SR < 2 RC, or SO < 3 RC,
that is, CR:CS>1:3.
Again, MC:CR = BC:CM
> 45 : 1 ; (see Prop. 11)
therefore, ex aequali,
CM:CS> 15:1.
And BC:CM> 45:1;
therefore BC : CS > 6 7 5 : 1 .
Prop. 15 (Fig. 1).
We have XO : (diam. of sun) < 1 : 9, (Prop. 13)
and, a fortiori, YZ : NO > 9 : 1 ;
therefore, by similar triangles, if YO, ZN meet in X,
AX:XR>9:1,
and convertendo, XA :AR< 9:8.
But AB > 18 BC, and, a fortiori, > 18 BR,
whence AB > IS (AR-AB), or 19 AB > 18 AR;
that is, AR:AB < 19:18.
Therefore, ex aequali,
XA:AB < 19:16,
and, convertendo, AX : XB > 19:3;
therefore (diam. of sun) : (diam. of earth) > 19:3.
Lastly, since CB.CR > 675 : 1, (Prop. 14)
CB:BR< 675:674.
ARISTARCHUS OF SAMOS 15
But AB.BG < 20:1;
therefore, ex aequali,
AB.BR < 13500:674
< 6750:337,
whence, by inversion and componendo,
RA:AB > 7087:6750. (1)
But AX:XR=YZ:X0
< 10125:979; (Prop. 13)
therefore, convertendo,
XA:AR > 10125:9146.
From this and (1) we have, ex aequali,
XA :AB > 10125 X 7087:9146x6750
> 71755875 : 61735500
> 43 : 37, a fortiori.
[It is difficult not to see in 43:37 the expression 1 + ,
6+6
which suggests that 43 : 37 was obtained by developing the
ratio as a continued fraction.]
Therefore, convertendo,
XA.XB < 43:6,
whence (diam. of sun) : (diam. of earth) < 43 : 6. Q. E. D.
XIII
ARCHIMEDES
The siege and capture of Syracuse by Marcellus during the
second Punic war furnished the occasion for the appearance of
Archimedes as a personage in history ; it is with this histori-
cal event that most of the detailed stories of him are con-
nected ; and the fact that he was killed in the sack of the city
in 212 B.C., when he is supposed to have been 75 years of age,
enables us to fix his date at about 287-212 B.C. He was the
son of Phidias, the astronomer, and was on intimate terms
with, if not related to, King Hieron and his son Gelon. It
appears from a passage of Diodorus that he spent some time
in Egypt, which visit was the occasion of his discovery of the
so-called Archimedean screw as a means of pumping water. 1
It may be inferred that he studied at Alexandria with the
successors of Euclid. It was probably at Alexandria that he
made the acquaintance of Conon of Samos (for whom he had
the highest regard both as a mathematician and a friend) and
of Eratosthenes of Cyrene. To the former he was in the habit
of communicating his discourses before their publication ;
while it was to Eratosthenes that he sent The Method, with an
introductory letter which is of the highest interest, as well as
(if we may judge by its heading) the famous Cattle- Problem.
Traditions.
It is natural that history or legend should say more of his
mechanical inventions than of his mathematical achievements,
which would appeal less to the average mind. His machines
were used with great effect against the Romans in the siege
of Syracuse. Thus he contrived (so we are told) catapults so
ingeniously constructed as to be equally serviceable at long or
1 Diodorus, v. 37. 3.
TRADITIONS 17
short range, machines for discharging showers of missiles
through holes made in the walls, and others consisting of
long movable poles projecting beyond the walls which either
dropped heavy weights on the enemy's ships, or grappled
their prows by means of an iron hand or a beak like that of
a crane, then lifted them into the air and let them fall again. 1
Marcellus is said to have derided his own engineers with the
words, 'Shall we not make an end of fighting against this
geometrical Briareus who uses our ships like cups to ladle
water from the sea, drives off our sambuca ignominiously
with cudgel-blows, and by the multitude of missiles that he
hurls at us all at once outdoes the hundred-handed giants of
mythology ? ' ; but all to no purpose, for the Romans were in
such abject terror that, ' if they did but see a piece of rope
or wood projecting above the wall, they would cry " there it
is", declaring that Archimedes was setting some engine in
motion against them, and would turn their backs and run
away '. 2 These things, however, were merely the * diversions
of geometry at play ', 3 and Archimedes himself attached no
importance to them. According to Plutarch,
' though these inventions had obtained for him the renown of
more than human sagacity, he yet would not even deign to
leave behind him any written work on such subjects, but,
regarding as ignoble and sordid the business of mechanics and
every sort of art which is directed to use and profit, he* placed
his whole ambition in those speculations the beauty and
subtlety of which is untainted by any admixture of the com-
mon needs of life.' 4
(a) Astronomy.
Archimedes did indeed write one mechanical book, On
Sphere-making, which is lost ; this described the construction
of a sphere to imitate the motions of the sun, moon and
planets. 5 Cicero saw this contrivance and gives a description
of it ; he says that it represented the periods of the moon
and the apparent motion of the sun with such accuracy that
it would even (over a short period) show the eclipses of the
sun and moon. 6 As Pappus speaks of ' those who understand
1 Polybius, Hist. viii. 7, 8 ; Livy xxiv. 34 ; Plutarch, Marcellus, cc. 15-17.
2 lb., c. 17. 3 lb., c. 14. , 4 lb., c. 17.
5 Carpus in Pappus, viii, p. 1026. 9 ; Proclus on Eucl. I, p. 41. 16.
6 Cicero, Be rep. i. 21, 22, Tusc. i. 63, Be naU deor. ii. 88.
1523.2 Q
18 ARCHIMEDES
the making of spheres and produce a, model of the heavens by
means of the circular motion of water', it is possible that
Archimedes's sphere was moved by water. In any case Archi-
medes was much occupied with astronomy. Livy calls him
' unicus spectator caeli siderumque '} Hipparchus says, ' From
these observations it is clear that the differences in the years
are altogether small, but, as to the solstices, I almost think
that Archimedes and I have both erred to the extent of a
quarter of a day both in the observation and in the deduction
therefrom \ 2 Archimedes then had evidently considered the
length of the year. Macrobius says he discovered the dis-
tances of the planets, 3 and he himself describes in his Sand-
reckoner the apparatus by which he measured the apparent
angular diameter of the sun.
(f3) Mechanics.
Archimedes wrote, as we shall see, on theoretical mechanics,
and it was by theory that he solved the problem To move a
given weight by a given force, for it was in reliance ' on the
irresistible cogency of his proof ' that he declared to Hieron
that any given weight could be moved by any given force
(however small), and boasted that, ' if he were given a place to
stand on, he could move the earth ' (ttol /?co, koX kli/co tolv yav,
as he said in his Doric dialect). The story, told by Plutarch,
is that, ' when Hieron was struck with amaze ment and asked
Archimedes to reduce the problem to practice and to give an
illustration of some great weight moved by a small force, he
fixed upon a ship of burden with three masts from the king's
arsenal which had only been drawn up with great labour by
many men, and loading her with many passengers and a full
freight, himself the while sitting far off, with no great effort
but only holding the end of a compound pulley (ttoXvo-ttckttos)
quietly in his hand and pulling at it, he drew the ship along
smoothly and safely as if she were moving through the sea.' 4
The story that Archimedes set the Roman ships on fire by
an arrangement of burning-glasses or concave mirrors is not
found in any authority earlier than Lucian; but it is quite
1 Livy xxiv. 34. 2. 2 Ptolemy, Syntaxis, III. 1, vol. i, p. 194. 23.
3 Macrobius, In Somn. Scip. ii. 3 ; cf. the figures in Hippolytus, Refut.,
p. 66. 52 sq., ed. Duncker.
4 Plutarch, Marcellus, c. 14.
MECHANICS 19
likely that he discovered some form of burning-mirror, e.g. a
paraboloid of revolution, which would reflect to one point all
rays falling on its concave surface in a direction parallel to
its axis.
Archimedes's own view of the relative importance of his
many discoveries is well shown by his request to his friends
and relatives that they should place upon his tomb a represen-
tation of a cylinder circumscribing a sphere, with an inscrip-
tion giving the ratio which the cylinder bears to the sphere ;
from which we may infer that he regarded the discovery of
this ratio as his greatest achievement. Cicero, when quaestor
in Sicily, found the tomb in a neglected state and repaired it x ;
but it has now disappeared, and no one knows where he was
buried.
Archimedes's entire preoccupation by his abstract studies is
illustrated by a number of stories. We are told that he would
forget all about his food and such necessities of life, and would
be drawing geometrical figures in the ashes of the fire or, when
anointing himself, in the oil on his body. 2 Of the same sort
is the tale that, when he discovered in a bath the solution of
the question referred to him by Hieron, as to whether a certain
crown supposed to have been made of gold did not in fact con-
tain a certain proportion of silver, he ran naked through the
street to his home shouting evprjKa, evprjKa. 3 He was killed
in the sack of Syracuse by a Roman soldier. The story is
told in various forms ; the most picturesque is that found in
Tzetzes, which represents him as saying to a Roman soldier
who found him intent on some diagrams which he had drawn
in the dust and came too close, ' Stand away, fellow, from my
diagram', whereat the man was so enraged that he killed
him. 4
Summary of main achievements.
In geometry Archimedes's work consists in the main of
original investigations into the quadrature of curvilinear
plane figures and the quadrature and cubature of curved
surfaces. These investigations, beginning where Euclid's
Book XII left off, actually (in the words of Chasles) ' gave
1 Cicero, Tusc. v. 64 sq. 2 Plutarch, Marcellus, c. 17,
3 Vitruvius, De architectural ix. 1. 9, 10.
4 Tzetzes, Chiliad, ii. 35. 135.
c 2
20 ARCHIMEDES
birth to the calculus of the infinite conceived and brought to
perfection successively by Kepler, Cavalieri, Fermat, Leibniz
and Newton '. ' He performed in fact what is equivalent to
integration in finding the area of a parabolic segment, and of
a spiral, the surface and volume of a sphere and a segment of
a sphere, and the volumes of any segments of the solids of
revolution of the second degree. In arithmetic he calculated
approximations to the value of 77-, in the course of which cal-
culation he shows that he could approximate to the value of
square roots of large or small non-square numbers ; he further
invented a system of arithmetical terminology by which he
could express in language any number up to that which we
should write down with 1 followed by 80,000 million million
ciphers. In mechanics he not only worked out the principles of
the subject but advanced so far as to find the centre of gravity
of a segment of a parabola, a semicircle, a cone, a hemisphere,
a segment of a sphere, a right segment of a paraboloid and
a spheroid of revolution. His mechanics, as we shall see, has
become more important in relation to his geometry since the
discovery of the treatise called The Method which was formerly
supposed to be lost. Lastly, he invented the whole science of
hydrostatics, which again he carried so far as to give a most
complete investigation of the positions of rest and stability of
a right segment of a paraboloid of revolution floating in a
fluid with its base either upwards or downwards, but so that
the base is either wholly above or wholly below the surface of m
the fluid. This represents a sum of mathematical achieve-
ment unsurpassed by any one man in the world's history.
Character of treatises.
The treatises are, without exception, monuments of mathe-
matical exposition ; the gradual revelation of the plan of
attack, the masterly ordering of the propositions, the stern
elimination of everything not immediately relevant to the
purpose, the finish of the whole, are so impressive in their
perfection as to create a feeling akin to awe in the mind of
the reader. As Plutarch said, ' It is not possible to find in
geometry more difficult and troublesome questions or proofs
set out in simpler and clearer propositions '} There is at the
1 Plutarch, Marcellus, c. 17.
CHARACTER OF TREATISES 21
same time a certain mystery veiling the way in which he
arrived at his results. For it is clear that they were not
discovered by the steps which lead up to them in the finished
treatises. If the geometrical treatises stood alone, Archi-
medes might seem, as Wallis said, ' as it were of set purpose
to have covered up the traces of his investigation, as if he had
grudged posterity the secret of his method of inquiry, while
he wished to extort from them assent to his results \ And
indeed (again in the words of Wallis) ' not only Archimedes
but nearly all the ancients so hid from posterity their method
of Analysis (though it is clear that they had one) that more
modern mathematicians found it easier to invent a new
Analysis than to seek out the old'. A partial exception is
now furnished by The Method of Archimedes, so happily dis-
covered by Heiberg. In this book Archimedes tells us how
he discovered certain theorems in quadrature and cubature,
namely by the use of mechanics, weighing elements of a
figure against elements of another simpler figure the mensura-
tion of which was already known. At the same time he is
careful to insist on the difference between (1) the means
which may be sufficient to suggest the truth of theorems,
although not furnishing scientific proofs of them, and (2) the
rigorous demonstrations of them by orthodox geometrical
methods which must follow before they can be finally accepted
as established :
' certain things ', he says, ' first became clear to me by a
mechanical method, although they had to be demonstrated by
geometry afterwards because their investigation by the said
method did not furnish an actual demonstration. But it is
of course easier, when we have previously acquired, by the
method, some knowledge of the questions, to supply the proof
than it is to find it without any previous knowledge.' ' This ',
he adds, ' is a reason why, in the case of the theorems that
the volumes of a cone and a pyramid are one-third of the
volumes of the cylinder and prism respectively having the
same base and equal height, the proofs of which Eudoxus was
the first to discover, no small share of the credit should be
given to Democritus who was the first to state the fact,
though without proof.'
Finally, he says that the very first theorem which he found
out by means of mechanics was that of the separate treatise
22 ARCHIMEDES
on the Quadrature of the parabola, namely that the area of any
segment of a section of a right-angled cone (i. e. a parabola) is
four-thirds of that of the triangle which has the same base and
height. The mechanical proof, however, of this theorem in the
Quadrature of the Parabola is different from that in the
Method, and is more complete in that the argument is clinched
by formally applying the method of exhaustion.
List of works still extant.
The extant works of Archimedes in the order in which they
appear in Heiberg's second edition, following the order of the
manuscripts so far as the first seven treatises are concerned,
are as follows :
(5) On the Sphere and Cylinder : two Books.
(9) Measurement of a Circle.
(7) On Conoids and Si^heroids.
(6) On Sirirals.
(1) On Plane Equilibriums, Book I.
(3) „ „ „ Book II.
(10) The Sand-reckoner (Psammites).
(2) Quadrature of the Parabola.
(8) On Floating Bodies: two Books.
? Stomachion (a fragment).
(4) The Method.
This, however, was not the order of composition ; and,
judging (a) by statements in Archimedes's own prefaces to
certain of the treatises and (6) by the use in certain treatises
of results obtained in others, we can make out an approxi-
mate chronological order, which I have indicated in the above
list by figures in brackets. The treatise On Floating Bodies
was formerly only known in the Latin translation by William
of Moerbeke, but the Greek text of it has now been in great
part restored by Heiberg from the Constantinople manuscript
which also contains The Method and the fragment of the
Stomachion.
Besides these works we have a collection of propositions
(Liber assumptorum) which has reached us through the
Arabic. Although in the title of the translation by Thabit b.
LIST OF EXTANT WORKS 23
Qurra the book is attributed to Archimedes, the propositions
cannot be his in their present form, since his name is several
times mentioned in them ; but it is quite likely that some
of them are of Archimedean origin, notably those about the
geometrical figures called apfirjXos (' shoemaker's knife ') and
aakivov (probably ' salt-cellar ') respectively and Prop. 8 bear-
ing on the trisection of an angle.
There is also the Cattle- Problem in epigrammatic form,
which purports by its heading to have been communicated by
Archimedes to the mathematicians at Alexandria in a letter
to Eratosthenes. Whether the epigrammatic form is due to
Archimedes himself or not, there is no sufficient reason for
doubting the possibility that the substance of it was set as a
problem by Archimedes.
Traces of lost works.
Of works which are lost we have the following traces.
1. Investigations relating to polyhedra are referred to by
Pappus who, after alluding to the five regular polyhedra,
describes thirteen others discovered by Archimedes which are
semi-regular, being contained by polygons equilateral and
equiangular but not all similar. 1
2. There was a book of arithmetical content dedicated to
Zeuxippus. We learn from Archimedes himself that it dealt
with the naming of numbers (/caro^o/za^y rcou dpid/icou) 2 and
expounded the system, which we find in the Sand-reckoner, of
expressing numbers higher than those which could be written
in the ordinary Greek notation, numbers in fact (as we have
said) up to the enormous figure represented by 1 followed by
80,000 million million ciphers.
3. One or more works on mechanics are alluded to contain-
ing propositions not included in the extant treatise On Plane
Equilibriums. Pappus mentions a work On Balances or Levers
(wepl (vycov) in which it was proved (as it also was in Philon's
and Heron's Mechanics) that ' greater circles overpower lesser
circles when they revolve about the same centre '. 3 Heron, too,
speaks of writings of Archimedes ' which bear the title of
1 Pappus, v, pp. 352-8.
2 Archimedes, vol. ii, pp. 216. 18, 236. 17-22 ; ef. p. 220. 4.
3 Pappus, viii, p. 1068.
24 • ARCHIMEDES
" works on the lever " \ l Simplicius refers to problems on the
centre of gravity, KevrpofiapiKci, such as the many elegant
problems solved by Archimedes and others, the object of which
is to show how to find the centre of gravity, that is, the point
in a body such that if the body is hung up from it, the body
will remain at rest in any position. 2 This recalls the assump-
tion in the Quadrature of the Parabola (6) that, if a body hangs
at rest from a point, the centre of gravity of the body and the
point of suspension are in the same vertical line. Pappus has
a similar remark with reference to a point of support, adding
that the centre of gravity is determined as the intersection of
two straight lines in the body, through two points of support,
which straight lines are vertical when the body is in equilibrium
so supported. Pappus also gives the characteristic of the centre
of gravity mentioned by Simplicius, observing that this is
the most fundamental principle of the theory of the centre of
gravity, the elementary propositions of which are found in
Archimedes's On Equilibriums (nepl io-oppoiricov) and Heron's
Mechanics. Archimedes himself cites propositions which must
have been proved elsewhere, e. g. that the centre of gravity
of a Cone divides the axis in the ratio 3:1, the longer segment
being that adjacent to the vertex 3 ; he also says that ' it is
proved in the Equilibriums ' that the centre of gravity of any
segment of a right-angled conoid (i. e. paraboloid of revolution)
divides the axis in such a way that the portion towards the
vertex is double of the remainder. 4 It is possible that there
was originally a larger work by Archimedes On Equilibriums
of which the surviving books On Plane Equilibriums formed
only a part ; in that case irepl £vyS>v and KevrpofiapiKoi may
only be alternative titles. Finally, Heron says that Archi-
medes laid down a certain procedure in a book bearing the
title ' Book on Supports \ 6
4. Theon of Alexandria quotes a proposition from a work
of Archimedes called Catoptrica (properties of mirrors) to the
effect that things thrown into water look larger and still
larger the farther they sink. 6 Olympiodorus, too, mentions
1 Heron, Mechanics, i. 32.
2 Simpl. on Arist. Be caelo, ii, p. 508 a 30, Brandis ; p. 543. 24, Heib.
3 Method, Lemma 10. 4 On Floating Bodies, ii. 2.
5 Heron, Mechanics, i. 25.
6 Theon on Ptolemy's Syntaxis, \, p. 29, Halma.
TRACES OF LOST WORKS 25
that Archimedes proved the phenomenon of refraction ' by
means of the ring placed in the vessel (of water) '- 1 A scholiast
to the Pseudo-Euclid's Catoptrica quotes a proof, which he
attributes to Archimedes, of the equality of the angles of
incidence and of reflection in a mirror.
The text of Archimedes.
Heron, Pappus and Theon all cite works of Archimedes
which no longer survive, a fact which shows that such works
were still extant at Alexandria as late as the third and fourth
centuries a.d. But it is evident that attention came to be
concentrated on two works only, the Measurement of a Circle
and On the Sphere and Cylinder. Eutocius {jl. about a.d. 500)
only wrote commentaries on these works and on the Plane
Equilibriums, and he does not seem even to have been
acquainted with the Quadrature of the Parabola or the work
On Spirals, although these have survived. Isidorus of Miletus
revised the commentaries of Eutocius on the Measurement
of a Circle and the two Books On tlie Sphere and Cylinder,
and it would seem to have been in the school of Isidorus
that these treatises were turned from their original Doric
into the ordinary language, with alterations designed to make
them more intelligible to elementary pupils. But neither in
Isidorus's time nor earlier was there any collected edition
of Archimedes's works, so that those which were less read
tended to disappear.
In the ninth century Leon, who restored the University
of Constantinople, collected together all the works that he
could find at Constantinople, and had the manuscript written
(the archetype, Heiberg's A) which, through its derivatives,
was, up to the discovery of the Constantinople manuscript (C)
containing The Method, the only source for the Greek text.
Leon's manuscript came, in the twelfth century, to the
Norman Court at Palermo, and thence passed to the House
of Hohenstaufen. Then; with all the library of Manfred, it
was given to the Pope by Charles of Anjou after the battle
of Benevento in 1266. It was in the Papal Library in the
years 1269 and 1311, but, some time after 1368, passed into
1 Olympiodorus on Arist. Meteorologica, ii, p. 94, Ideler ; p. 211. 18,
Busse.
26 ARCHIMEDES
private hands. In 1491 it belonged to Georgius Valla, who
translated from it the portions published in his posthumous
work De expetendis et fugiendis rebus (1501), and intended to
publish the whole of Archimedes with Eutocius's commen-
taries. On Valla's death in 1500 it was bought by Albertus-
Pius, Prince of Carpi, passing in 1530 to his nephew, Rodolphus
Pius, in whose possession it remained till 1544. At some
time between 1544 and 1564 it disappeared, leaving no
trace.
The greater part of A was translated into Latin in 1269
by William of Moerbeke at the Papal Court at Viterbo. This
translation, in William's own hand, exists at Rome (Cod.
Ottobon. lat. 1850, Heiberg's B), and is one of our prime
sources, for, although the translation was hastily done and
the translator sometimes misunderstood the Greek, he followed
its wording so closely that his version is, for purposes of
collation, as good as a Greek manuscript. William used also,
for his translation, another manuscript from the same library
which contained works not included in A. This manuscript
was a collection of works on mechanics and optics ; William
translated from it the two Books On Floating Bodies, and it
also contained the Plane Equilibriums and the Quadrature
of the Parabola, for which books William used both manu-
scripts.
The four most important extant Greek manuscripts (except
C, the Constantinople manuscript discovered in 1906) were
copied from A. The earliest is E, the Venice manuscript
(Marcianus 305), which was written between the years 1449
and 1472. The next is D, the Florence manuscript (Laurent.
XXVIII. 4), which was copied in 1491 for Angelo Poliziano,
permission having been obtained with some difficulty in con-
sequence of the jealousy with which Valla guarded his treasure.
The other two are G (Paris. 2360) copied from A after it had
passed to Albertus Pius, and H (Paris. 2361) copied in 1544
by Christopherus Auverus for Georges d'Armagnac, Bishop
of Rodez. These four manuscripts, with the translation of
William of Moerbeke (B), enable the readings of A to be
inferred. ,
A Latin translation was made at the instance of Pope
Nicholas V about the year 1450 by Jacobus Cremonensis.
THE TEXT OF ARCHIMEDES 27
It was made from A, which was therefore accessible to Pope
Nicholas though it does not seem to have belonged to him.
Regiomontanus made a copy of this translation about 1468
and revised it with the help of E (the Venice manuscript of
the Greek text) and a copy of the same translation belonging
to Cardinal Bessarion, as well as another ' old copy ' which
seems to have been B.
The editio princeps was published at Basel (apud Herva-
gium) by Thomas GechaufF Venatorius in 1544. The Greek
text was based on a Nurnberg MS. (Norimberg. Cent. V,
app. 12) which was copied in the sixteenth century from A
but with interpolations derived from B ; the Latin transla-
tion was Regiomontanus's revision of Jacobus Cremonensis
(Norimb. Cent. V, 15).
A translation by F. Commandinus published at Venice in
1558 contained the Measurement of a Circle, On Spirals, the
Quadrature of the Parabola, On Conoids and Spheroids, and
the Sand-reckoner. This translation was based- 1 on the Basel
edition, but Commandinus also consulted E and other Greek
manuscripts.
Torelli's edition (Oxford, 1792) also followed the editio
princeps in the main, but Torelli also collated E. The book
was brought out after Torelli's death by Abram Robertson,
who also collated five more manuscripts, including D, G
and H. The collation, however, was not well done, and the
edition was not properly corrected when in the press.
The second edition of Heiberg's text of all the works of
Archimedes with Eutocius's commentaries, Latin translation,
apparatus criticus, &c, is now available (1910-15) and, of
course, supersedes the first edition (1880-1) and all others.
It naturally includes The Method, the fragment of the Stoma-
chion, and so much of the Greek text of the two Books On
Floating Bodies as could be restored from the newly dis-
covered Constantinople manuscript. 1
Contents of The Method.
Our description of the extant works of Archimedes
may suitably begin with The Method (the full title is On
1 The Works of Archimedes, edited in modern notation by the present
writer in 1897, was based on Heiberg's first edition, and the Supplement
28 ARCHIMEDES
Mechanical Theorems, Method (communicated) to Eratosthenes).
Premising certain propositions in mechanics mostly taken
from the Plane Equilibriums, and a lemma which forms
Prop. 1 of On Conoids and Spheroids, Archimedes obtains by
his mechanical method the following results. The area of any
segment of a section of a right-angled cone (parabola) is § of
the triangle with the same base and height (Prop. 1). The
right cylinder circumscribing a sphere or a spheroid of revolu-
tion and with axis equal to the diameter or axis of revolution
of the sphere or spheroid is 1\ times the sphere or spheroid
respectively (Props. 2, 3). Props. 4, 7,8,11 find the volume of
any segment cut off, by a plane at right angles to the axis,
from any right-angled conoid (paraboloid of revolution),
sphere, spheroid, and obtuse-angled conoid (hyperboloid) in
terms of the cone which has the same base as the segment and
equal height. In Props. 5, 6, 9, 10 Archimedes uses his method
to find the centre of gravity of a segment of a paraboloid of
revolution, a sphere, and a spheroid respectively. Props.
12-15 and Prop. 16 are concerned with the cubature of two
special solid figures. (1) Suppose a prism with a square base
to have a cylinder inscribed in it, the circular bases of the
cylinder being circles inscribed in the squares which are
the bases of the prism, and suppose a plane drawn through
one side of one base of the prism and through that diameter of
the circle in the opposite base which is parallel to the said
side. This plane cuts off a solid bounded by two planes and
by part of the curved surface of the cylinder (a solid shaped
like a hoof cut off by a plane); and Props. 12-15 prove that
its volume is one-sixth of the volume of the prism. (2) Sup-
pose a cylinder inscribed in a cube, so that the circular bases
of the cylinder are circles inscribed in two opposite faces of
the cube, and suppose another cylinder similarly inscribed
with reference to two other opposite faces. The two cylinders
enclose a certain solid which is actually made up of eight
'hoofs' like that of Prop. 12. Prop. 16 proves that the
volume of this solid is two-thirds of that of the cube. Archi-
medes observes in his preface that a remarkable fact about
(1912) containing The Method, on the original edition of Heiberg (in
Hermes, xlii, 1907) with the translation by Zeuthen (Bibliotheca Mathe-
matical, vii s . 1906/7).
THE METHOD 29
these solids respectively is that each of them is equal to a
solid enclosed by planes, whereas the volume of curvilinear
solids (spheres, spheroids, &c.) is generally only expressible in
terms of other curvilinear solids (cones and cylinders). In
accordance with his dictum that the results obtained by the
mechanical method are merely indicated, but not actually
proved, unless confirmed by the rigorous methods of pure
geometry, Archimedes proved the facts about the two last-
named solids by the orthodox method of exhaustion as
regularly used by him in his other geometrical treatises ; the
proofs, partly lost, were given in Props. 15 and 16.
We will first illustrate the method by giving the argument
of Prop. 1 about the area of a parabolic segment.
Let ABO be the segment, BD its diameter, OF the tangent
at 0. Let P be any point on the segment, and let AKF,
OPNM be drawn parallel to BD. Join CB and produce it to
meet MO in N and FA in K, and let KH be made equal to
KG.
Now, by a proposition ' proved in a lemma ' (cf . Quadrature
of the Parabola, Prop. 5)
MO:OP= OA:A0
= CK:KN '
= HK:KN.
Also, by the property of the parabola, EB = BD, so that
MN = NO and FK = KA.
It follows that, if HO be regarded as the bar of a balance,
a line TG equal to PO and placed with its middle point at H
balances, about K, the straight line MO placed where it is,
i. e. with its middle point at N.
Similarly with all lines, as MO, PO, in the triangle GFA
and the segment CBA respectively.
And there are the same number of these lines. Therefore
30 ARCHIMEDES
the whole segment of the parabola acting at H balances the
triangle CFA placed where it is.
But the centre of gravity of the triangle CFA is at W,
where CW = 2 WK [and the whole triangle may be taken as
acting at W\
Therefore (segment ABC) : A CFA = WK : KH
= 1:3,
so that (segment ABC) = ±ACFA
= %AABC. Q.E.D.
It will be observed that Archimedes takes the segment and
the triangle to be made wp of parallel lines indefinitely close
together. In reality they are made up of indefinitely narrow
strips, but the width (dx, as we might say) being the same
for the elements of the triangle and segment respectively,
divides out. And of course the weight of each element in
both is proportional to the area. Archimedes also, without
mentioning moments, in effect assumes that the sum of the
moments of each particle of a figure, acting where it is, is
equal to the moment of the whole figure applied as one mass
at its centre of gravity.
We will now take the case of any segment of a spheroid
of revolution, because that will cover several propositions of
Archimedes as particular cases.
The ellipse with axes AA\ BB r is a section made by the
plane of the paper in a spheroid with axis A A'. It is required
to find the volume of any right segment ADC of the spheroid
in terms of the right cone with the same base and height.
Let DC be the diameter of the circular base of the segment.
Join AB, AB', and produce them to meet the tangent at A' to
the ellipse in K, K', and DC produced in E, F.
Conceive a cylinder described with axis AA f and base the
circle on KK f as diameter, and cones described with iff as
axis and bases the circles on EF, DC as diameters.
Let N be any point on AG, and let MOPQNQ'P'O'M' be
drawn through N parallel to BB' or DC as shown in the
figure.
Produce A' A to H so that HA = A A'.
Now
THE METHOD
HA:AN=A'A:AN
= KA:AQ
= MN:NQ
= MN 2 :MN.NQ.
It is now necessary to prove that MN.NQ = NP 2 + NQ 2 .
H
31
M
*t
A
p/q/7
\\qNp'
0'
B
m
N \\\
w \ V
B'
V \ ,
/e
V
G C/
\F
M'
K A ; K'
By the property of the ellipse,
AN. NA' : NP 2 = dAA') 2 : QBB') 2
= AN 2 :NQ 2 ;
therefore NQ 2 : NP 2 = AN 2 :AN. NA'
= NQ 2 :NQ.QM,
whence NP 2 = MQ . QN.
Add NQ 2 to each side, and we have
NP 2 + NQ 2 = MN.NQ.
Therefore, from above,
HA:AN= MN 2 : (NP 2 + NQ 2 ).
(1)
But MN 2 , NP 2 , NQ 2 are to one another as the areas of the
circles with MM', PP' ', QQ' respectively as diameters, and these
32 ARCHIMEDES
circles are sections made by the plane though iV at right
angles to A A' in the cylinder, the spheroid and the cone AEF
respectively.
Therefore, if HA A' be a lever, and the sections of the
spheroid and cone be both placed with their centres of gravity
at H, these sections placed at H balance, about A, the section
MM' of the cylinder where it is.
Treating all the corresponding sections of the segment of
the spheroid, the cone and the cylinder in the same way,
we find that the cylinder with axis AG, where it is, balances,
about A, the cone AEF and the segment ADC together, when
both are placed with their centres of gravity at H; and,
if W be the centre of gravity of the cylinder, i. e. the middle
point of AG,
HA :AW = (cylinder, axis AG) : (cone AEF+ segmt. ADC).
If we call V the volume of the cone AEF, and S that of the
segment of the spheroid, we have
A A' 2
(cylinder) : (V+S) = 37.^ : (V+S),
while HA:AW= A A' :\AG.
A A' 2
Therefore AA' \\AG = 3 V.-^- 2 : (V + S),
jOl(jT
A A'
and (V+S) = tV.£±,
whence 8 = V( — — - — 1 V
\2AG )
Again, let V be the volume of the cone A DC.
Then V:V'=EG 2 :DG 2
BB' 2
— .AG 2 DG 2
AA' 2 '
But DG 2 :AG.GA' = BB /2 :AA' 2 .
Therefore V: V = AG 2 : AG. G A'
= AG:GA\
THE METHOD 33
, AG/3AA'
It follows that S=V. ~^ f (jjq ~ l)
%AA'-AG
= V .
/ ^
A'G
= V.
, iAA' + A'G
A'G
which is the result stated by Archimedes in Prop. 8.
The result is the same for the segment of a sphere. The
proof, of course slightly simpler, is given in Prop. 7.
In the particular case where the segment is half the sphere %
or spheroid, the relation becomes
S = 2 V\ (Props. 2, 3)
and it follows that the volume of the whole sphere or spheroid
is 4 V\ where V is the volume of the cone ABB' \ i.e. the
volume of the sphere or spheroid is two-thirds of that of the
circumscribing cylinder.
In order now to find the centre of gravity of the segment
of a spheroid, we must have the segment acting where it is,
not at H.
Therefore formula (1) above will not serve. But we found
that MN . NQ = (i^P 2 + JVQ 2 ),
whence MJSf 2 : (iVT 2 + NQ 2 ) = (FP 2 + FQ 2 ) : NQ 2 ;
therefore HA : AN = (NP 2 + NQ 2 ) : NQ 2 .
(This is separately proved by Archimedes for the sphere
in Prop. 9.)
From this we derive, as usual, that the cone AEF and the
segment ADC both acting where they are balance a volume
equal to the cone A EF placed with its centre of gravity at H.
Now the centre of gravity of the cone AEF is on the line
A G at a distance f AG from A. Let X be the required centre
of gravity of the segment. Then, taking moments about A,
we have V .HA = S.AX+V.iAG,
or V(AA'-iAG) = S.AX
= y(^~rn l)AX y from above.
1523.2 D
34 ARCHIMEDES
Therefore AX: AG = (AA'-$AG) : (%AA'-AG)
= (4AA'-3AG):(6AA'-4AG);
whence AX:XG = (4AA'- 3AG) : (2AA'-AG)
= (AG + ±A'G):\AG + 2A'G),
which is the result obtained by Archimedes in Prop. 9 for the
sphere and in Prop. 10 for the spheroid.
In the case of the hemi-spheroid or hemisphere the ratio
AX : XG becomes 5 : 3, a result obtained for the hemisphere in
Prop. 6.
The cases of the paraboloid of revolution (Props. 4, 5) and
the hyperboloid of revolution (Prop. 11) follow the same course,
and it is unnecessary to reproduce them.
For the cases of the two solids dealt with at the end of the
treatise the reader must be referred to the propositions them-
selves. Incidentally, in Prop. 13, Archimedes finds the centre
of gravity of the half of a cylinder cut by a plane through
the axis, or, in other words, the centre of gravity of a semi-
circle.
We will now take the other treatises in the order in which
they appear in the editions.
On the Sphere and Cylinder, I, II.
The main results obtained in Book I are shortly stated in
a prefatory letter to Dositheus. Archimedes tells us that
they are new, and that he is now publishing them for the
first time, in order that mathematicians may be able to ex-
amine the proofs and judge of their value. The results are
(1) that the surface of a sphere is four times that of a great
circle of the sphere, (2) that the surface of any segment of a
sphere is equal to a circle the radius of which is equal to the
straight line drawn from the vertex of the segment to a point
on the circumference of the base, (3) that the volume of a
cylinder circumscribing a sphere and with height equal to the
diameter of the sphere is § of the volume of the sphere,
(4) that the surface of the circumscribing cylinder including
its bases is also § of the surface of the sphere. It is worthy
of note that, while the first and third of these propositions
appear in the book in this order (Props. 33 and 34 respec-
ON THE SPHERE AND CYLINDER, I 35
tively), this was not the order of their discovery ; for Archi-
medes tells us in The Method that
' from the theorem that a sphere is four times as great as the
cone with a great circle of the sphere as base and with height
equal to the radius of the sphere I conceived the notion that
the surface of any sphere is four times as great as a great
circle in it ; for, judging from the fact that any circle is equal
to a triangle with base equal to the circumference and height
equal to the radius of the circle, I apprehended that, in like
manner, any sphere is equal to a cone with base equal to the
surface of the sphere and height equal to the radius '.
Book I begins with definitions (of ' concave in the same
direction ' as applied to curves or broken lines and surfaces, of
a ' solid sector ' and a ' solid rhombus ') followed by five
Assumptions, all of importance. Of all lines ivhich have the
same extremities the straight line is the least, and, if there are
two curved or bent lines in a plane having the same extremi-
ties and concave in the same direction, but one is wholly
included by, or partly included by and partly common with,
the other, then that which is included is the lesser of the two.
Similarly with plane surfaces and surfaces concave in the
same direction. Lastly, Assumption 5 is the famous ' Axiom
of Archimedes ', which however was, according to Archimedes
himself, used by earlier geometers (Eudoxus in particular), to
the effect that Of unequal magnitudes the greater exceeds
the less by such a magnitude as, when added to itself, can be
made to exceed any assigned magnitude of the same kind ;
the axiom is of course practically equivalent to Eucl. V, Def. 4,
and is closely connected with the theorem of Eucl. X. 1.
As, in applying the method of exhaustion, Archimedes uses
both circumscribed and inscribed figures with a view to com-
pressing them into coalescence with the curvilinear figure to
be measured, he has to begin with propositions showing that,
given two unequal magnitudes, then, however near the ratio
of the greater to the less is to 1, it is possible to find two
straight lines such that the greater is to the less in a still less
ratio ( > 1), and to circumscribe and inscribe similar polygons to
a circle or sector such that the perimeter or the area of the
circumscribed polygon is to that of the inner in a ratio less
than the given ratio (Props. 2-6): also, just as Euclid proves
D 2
36 ARCHIMEDES
that, if we continually double the number of the sides of the
regular polygon inscribed in a circle, segments will ultimately be
left which are together less than any assigned area, Archimedes
has to supplement this (Prop. 6) by proving that, if we increase
the number of the sides of a circumscribed regular polygon
sufficiently, we can make the excess of the area of the polygon
over that of the circle less than any given area. Archimedes
then addresses himself to the problems of finding the surface of
any right cone or cylinder, problems finally solved in Props. 1 3
(the cylinder) and 14 (the cone). Circumscribing and inscrib-
ing regular polygons to the bases of the cone and cylinder, he
erects pyramids and prisms respectively on the polygons as
bases and circumscribed or inscribed to the cone and cylinder
respectively. In Props. 7 and 8 he finds the surface of the
pyramids inscribed and circumscribed to the cone, and in
Props. 9 and 10 he proves that the surfaces of the inscribed
and circumscribed pyramids respectively (excluding the base)
are less and greater than the surface of the cone (excluding
the base). Props. 11 and 12 prove the same thing of the
prisms inscribed and circumscribed to the cylinder, and finally
Props. 13 and 14 prove, by the method of exhaustion, that the
surface of the cone or cylinder (excluding the bases) is equal
to the circle the radius of which is a mean proportional
between the ' side ' (i. e. generator) of the cone or cylinder and
the radius or diameter of the base (i.e. is equal to wrs in the
case of the cone and 2irrs in the case of the cylinder, where
r is the radius of the base and s a generator). As Archimedes
here applies the method of exhaustion for the first time, we
will illustrate by the case of the cone (Prop. 14). .
Let A be the base of the cone, C a straight line equal to its
c
E
radius, D a line equal to a generator of the cone, E a mean
proportional to G, D, and B a circle with radius equal to E.
ON THE SPHERE AND CYLINDER, I 37
If $ is the surface of the cone, we have to prove that S= B.
For, if S is not equal to B, it must be either greater or less.
I. Suppose B < S.
Circumscribe a regular polygon about B, and inscribe a similar
polygon in it, such that the former has to the latter a ratio less
than S:B (Prop. 5). Describe about A a similar polygon and
set up from it a pyramid circumscribing the cone.
Then (polygon about A) : (polygon about B)
= C 2 :E 2
= C:D
— (polygon about A) : (surface of pyramid).
Therefore (surface of pyramid) = (polygon about B).
But (polygon about B) : (polygon in B) < S : B ;
therefore (surface of pyramid) : (polygou in B) < S : B.
But this is impossible, since (surface of pyramid) > S, while
(polygon in B) < B;
therefore B is not less than S.
II. Suppose B > S,
Circumscribe and inscribe similar regular polygons to B
such that the former has to the latter a ratio < B : S. Inscribe
in J. a similar polygon, and erect on A the inscribed pyramid.
Then (polygon in A) : (polygon in B) — C 2 : E 2
= C:D
> (polygon in A) : (surface of pyramid).
(The latter inference is clear, because the ratio of C:D is
greater than the ratio of the perpendiculars from the centre of
A and from the vertex of the pyramid respectively on any
side of the polygon in A ; in other words, if /? < oc < \ir,
sincx > sin/?.)
Therefore (surface of pyramid) > (polygon in B).
But (polygon about B) : (polygon in B) < B : S,
whence (a fortiori)
(polygon about B) : (surface of pyramid) < B: S,
which is impossible, for (polygon about B) > B, while (surface
of pyramid) < #.
38
ARCHIMEDES
Therefore B is not greater than S.
Hence S, being neither greater nor less than B, is equal to B.
Archimedes next addresses himself to the problem of finding
the surface and volume of a sphere or a segment thereof, but
has to interpolate some propositions about ' solid rhombi '
(figures made up of two right cones, unequal or equal, with
bases coincident and vertices in opposite directions) the neces-
sity of which will shortly appear.
Taking a great circle of the sphere or a segment of it, he
inscribes a regular polygon of an even number of sides bisected
by the diameter A A', and approximates to the surface and
volume of the sphere or segment by making the polygon
revolve about A A' and measuring the surface and volume of
solid so inscribed (Props. 21-7). He then does the same for the
a circumscribed solid (Props. 28-32). Construct the inscribed
polygons as shown in the above figures. Joining BB\ CC\ ... ,
CB\ DC ... we see that BB\ CC ... are all parallel, and so are
AB,CB', DC ....
Therefore, by similar triangles, BF.FA — A'BiBA, and
BF.FA =B'F:FK
= CG:GK
= CG : GL
= E'I:IA' in Fig. 1
(= PM-.MN in Fig. 2),
ON THE SPHERE AND CYLINDER, I 39
whence, adding antecedents and consequents, we have
(Fig. 1) (BB' + 00'+... + EE') : A A' = A'B : BA, (Prop. 21)
(Fig. 2) {BR + CC+...+ \PP') :AM=A'B:BA. (Prop. 2 2)
When we make the polygon revolve about A A', the surface
of the inscribed figure so obtained is made up of the surfaces
of cones and frusta of cones; Prop. 14 has proved that the
surface of the cone ABB' is what we should write tt . AB . BF,
and Prop. 16 has proved that the surface of the frustum
BOC'B' is tt.BC(BF+GG). It follows that, since AB =
BG = . . . , the surface of the inscribed solid is
tt .AB {%BW + ^(BB' + GC')+ ...},
that is, tt . AB (BB' + 00'+... + EE') (Fig. 1), (Prop. 24)
or tt.AB (BB' + CC+...+ ±PP') (Fig. 2). (Prop. 35)
Hence, from above, the surface of the inscribed solid is
n . A'B . AA' or tt . A'B .AM, and is therefore less than
7T . AA' 2 (Prop. 25) or tt . A' A . AM, that is, tt . AP 2 (Prop. 37).
Similar propositions with regard to surfaces formed by the
revolution about AA' of regular circumscribed solids prove
that their surfaces are greater than it. A A' 2, and tt .AP 2
respectively (Props. 28-30 and Props. 39-40). The case of the
segment is more complicated because the circumscribed poly-
gon with its sides parallel to AB, BG ... DP circumscribes
the sector POP'. Consequently, if the segment is less than a
semicircle, as GAG', the base of the circumscribed polygon
(cc') is on the side of GC' towards A, and therefore the circum-
scribed polygon leaves over a small strip of the inscribed. This
complication is dealt with in Props. 39-40. Having then
arrived at circumscribed and inscribed figures with surfaces
greater and less than tt. AA' 2 and tt. AP 2 respectively, and
having proved (Props. 32, 41) that the surfaces of the circum-
scribed and inscribed figures are to one another in the duplicate
ratio of their sides, Archimedes proceeds to prove formally, by
the method of exhaustion, that the surfaces of the sphere and
segment are equal to these circles respectively (Props. 33 and
42); tt .AA' 2 is of course equal to four times the great circle
of the sphere. The segment is, for convenience, taken to be
40 ARCHIMEDES
less than a hemisphere, and Prop. 43 proves that the same
formula applies also to a segment greater than a hemisphere.
As regards the volumes different considerations involving
' solid rhombi ' come in. For convenience Archimedes takes,
in the case of the whole sphere, an inscribed polygon of 4n
sides (Fig. 1). It is easily seen that the solid figure formed
by its revolution is made up of the following : first, the solid
rhombus formed by the revolution of the quadrilateral AB0B f
(the volume of this is shown to be equal to the cone with base
equal to the surface of the cone ABB' and height equal to p,
the perpendicular from on AB, Prop. 18); secondly, the
extinguisher-shaped figure formed by the revolution of the
triangle BOG about A A' (this figure is equal to the difference
between two solid rhombi formed by the revolution of TBOB'
and TCOC respectively about A A', where T is the point of
intersection of GB, G'B' produced with A' A produced, and
this difference is proved to be equal to a cone with base equal
to the surface of the frustum of a cone described by BG in its
revolution and height equal to p the perpendicular from on
BG, Prop. 20) ; and so on ; finally, the figure formed by the
revolution of the triangle GOD about A A' is the difference
between a cone and a solid rhombus, which is proved equal to
a cone with base equal to the surface of the frustum of a cone
described by CD in its revolution and height p (Prop. 19).
Consequently, by addition, the volume of the whole solid of
revolution is equal to the cone with base equal to its whole
surface and height p (Prop. 26). But the whole of the surface
of the solid is less than 4 nr 2 , and p<r; therefore the volume
of the inscribed solid is less than four times the cone with
base nr 2 and height r (Prop. 27).
It is then proved in a similar way that the revolution of
the similar circumscribed polygon of 4^ sides gives a solid
the volume of which is greater than four times the same cone
(Props. 28-31 Cor.). Lastly, the volumes of the circumscribed
and inscribed figures are to one another in the triplicate ratio of
their sides (Prop. 32) ; and Archimedes is now in a position to
apply the method of exhaustion to prove that the volume of
the sphere is 4 times the cone with base nr 2 and height r
(Prop. 34).
Dealing with the segment of a sphere, Archimedes takes, for
ON THE SPHERE AND CYLINDER, I 41
convenience, a segment less than a hemisphere and, by the
same chain of argument (Props. 38, 40 Corr., 41 and 42), proves
(Prop. 44) that the volume of the sector of the sphere bounded
by the surface of the segment is equal to a cone with base
equal to the surface of the segment and height equal to the
radius, i. e. the cone with base w . AP 2 and height r (Fig. 2).
It is noteworthy that the proportions obtained in Props. 21,
22 (see p. 39 above) can be expressed in trigonometrical form.
If 4?i is the number of the sides of the polygon inscribed in
the circle, and 2n the number of the sides of the polygon
inscribed in the segment, and if the angle AOP is denoted
by a, the trigonometrical equivalents of the proportions are
respectively
IT « IT • 77" 7T
(1) sin f-sin- h ... +sm(2?i— 1) -— = cot — ;
w 2n 2n v ' 2n 4n
( . oc . 2oc . . .oc)
(2) 2 -J sin - +sin h ... + sm in — \)-\ + sina
In n n)
= (1 — cos oc) cot
v ' 2n
Thus the two proportions give in effect a summation of the
series
sin + sin 2 6 + . . . + sin (n — 1) 0,
both generally where nO is equal to any angle oc less than n
and in the particular case where n is even and 6 = ir / n.
Props. 24 and 35 prove that the areas of the circles equal to
the surfaces of the solids of revolution described by the
polygons inscribed in the sphere and segment are the above
series multiplied by Inr 2 sin — and nr 2 . 2 sin — respectively
4 n & n
TT oc
and are therefore 4 77- r 2 cos — and n r 2 . 2 cos — (1— cos a)
4 n 2n '
respectively. Archimedes's results for the surfaces of the
sphere and segment, 47rr 2 and 27rr 2 (l — cos a), are the
limiting values of these expressions when n is indefinitely
increased and when therefore cos — and cos — become
4ti 2n
unity. And the two series multiplied by 4 77- r 2 sin— and
In
42
ARCHIMEDES
~2n
ttt 2 .2 sin -^- respectively are (when n is indefinitely increased)
precisely what we should represent by the integrals
47rr 2 .-!
sin 6 dO, or 47rr 2 ,
and
7rr w .
2sin0cZ0, or 2irr 2 (l — cos a).
Book II contains six problems and three theorems. Of the
theorems Prop. 2 completes the investigation of the volume of
any segment of a sphere, Prop. 44 of Book I having only
brought us to the volume of the corresponding sector. If
ABB' be a segment of a sphere cut off by a plane at right
angles to AA', we learnt in I. 44 that the volume of the sector
OBAB' is equal to the cone with base equal to the surface
of the segment and height equal to the radius, i.e. \n . AB 2 .r,
where r is the radius. The volume of the segment is therefore
iTr.ABKr-iTr.BMKOM.
Archimedes wishes to express this as a cone with base the
same as that of the segment. Let AM, the height of the seg-
ment, = h.
Now AB 2 : BM 2 = A' A : A'M = 2r:(2r-h).
Therefore
in(AB 2 .r-BMKOM)=lw.BM 2 ^^-j i -(r-h)l
= i7r.BM 2 .h(^h.
3 v 2r — V
That is, the segment is equal to the cone with the same
base as that of the segment and height h(3r—h)/(2r — h).
ON THE SPHERE AND CYLINDER, II 43
This is expressed by Archimedes thus. If HM is the height
of the required cone,
HM:AM = (OA' + A'M):A'M, (1)
and similarly the cone equal to the segment A'BB' has the
height HM, where
HM : A'M = (OA + AM) : AM. (2)
His proof is, of course, not in the above form but purely
geometrical.
This proposition leads to the most important proposition in
the Book, Prop. 4, which solves the problem To cut a given
sphere by a plane in such a vjay that the volumes of the
segments are to one another in a given ratio.
Cubic equation arising out of II. 4.
If m : 7i be the given ratio of the cones which are equal to
the segments and the heights of which are h, h' , we have
Sr — h\ m , , /Zr — h' s
,/dr — /i\ _ m ,, /Sr — h, \
\2r — h) ' ' "n \2r — h')
and, if we eliminate h' by means of the relation h + h! — 2r,
we easily obtain the following cubic equation in h,
tf-3h 2 r+ -A— r 3 = 0.
m + n
Archimedes in effect reduces the problem to this equation,
which, however, he treats as a particular case of the more
general problem corresponding to the equation
(r + h):b = c 2 :(2r-h) 2 ,
where b is a given length and c 2 any given area,
or x 2 (a — x) = be 2 , where x = 2r—h and 3r = a.
Archimedes obtains his cubic equation with one unknown
by means of a geometrical elimination of H, H' from the
equation HM = — . R'M, where HM, HM have the values
n
determined by the proportions (1) and (2) above , after which
the one variable point M remaining corresponds to the one
unknown of the cubic equation. His method is, first, to find
44 ARCHIMEDES
values for each of the ratios A'H' : H'M and H'H: A'H' which
are alike independent of H, H' and then, secondly, to equate
the ratio compounded of these two to the known value of the
ratio HH' : H'M.
(ex.) We have, from (2),
A'H : H'M = OA : (OA + AM). (3)
(/?) From (1) and (2), separando,
AH:A3I= OA'iA'M, (4)
A'H' : A'M =0 A: AM. (5)
Equating the values of the ratio A'M : AM given by (4). (5),
we have 6 A' : AH = A'H' : OA -
= OH' : OH,
whence HH : OH = OH' : A'H', (since OA = OA')
or HH'.A'H' = OH 2 ,
so that HH' : A'H' = OH' 2 : ^i?' 2 . (6)
But, by (5), OA' : A'H' = AM: A'M,
and, componendo, OH : A'H' — AA' : A'M.
By substitution in (6),
HH' : A'H = A A' 2 : A'M 2 . (7)
Compounding with (3), we obtain
HH : H'M = (A A' 2 : A 'M 2 ) . (OA : OA + AM). (8)
[The algebraical equivalent of this is
m + n 4 r 3
n " (2r—h) 2 (r-{-h) i
... , . m + n 4r 3
which reduces to = ■—=-= =-z ,
on 3/rr — h 6
or h z — 3h 2 r-\ r 3 = 0, as above.]
m + n
Archimedes expresses the result (8) more simply by pro-
ducing OA to D so that OA = AD, and then dividing AD at
ON THE SPHERE AND CYLINDER, II 45
E so that AD:DE= HH'-.H'M or (m + n):n. We have
then OA = A D and OA+AM = MD, so that (8) reduces to
AD:DE = (A A'* : A'M 1 ) . (AD : MD),
or MD : DE = A A' 2 : A'M 2 .
Now, says Archimedes, D is given, since AD = OA. Also,
AD : DE being a given ratio, DE is given. Hence the pro-
blem reduces itself to that of dividing A'D into two parts at
M such that
MD : (a given length) = (a given area) : A'M 2 .
That is, the generalized equation is of the form
x 2 (a — x) = be 2 , as above.
(i) Archimedes's own solution of the cubic.
Archimedes adds that, ' if the problem is propounded in this
general form, it requires a Siopicr/ios [i.e. it is necessary to
investigate the limits of possibility], but if the conditions are
added which exist in the present case [i.e. in the actual
problem of Prop. 4], it does not require a Siopta-fios' (in other
words, a solution is always possible). He then promises to
give ' at the end ' an analysis and synthesis of both problems
[i.e. the Sioptcrfjios and the problem itself]. The promised
solutions do not appear in the treatise as we have it, but
Eutocius gives solutions taken from ' an old book ' which he
managed to discover after laborious search, and which, since it
was partly written in Archimedes's favourite Doric, he with
fair reason assumed to contain the missing addendum by
Archimedes.
In the Archimedean fragment preserved by Eutocius the
above equation, x 2 (a — x) = be 2 , is solved by means of the inter-
section of a parabola and a rectangular hyperbola, the equations
of which may be written thus
e 2
x 2 = — y, (a—x) y — ab.
a
The Siopio-fios takes the form of investigating the maximum
possible value of x 2 (a — x), and it is proved that this maximum
value for a real solution is that corresponding to the value
x = §a. This is established by showing that, if be 2 = -g^a?,
46
ARCHIMEDES
the curves touch at the point for which x = §a. If on the
other hand be 2 < -^fW\ it is proved that there are two real
solutions. In the particular case arising in Prop. 4 it is clear
that the condition for a real solution is satisfied, for the
m
expression corresponding to be 2 is 4r 3 , and it is only
m + n
necessary that
m
4r 3 should be not greater than ^Wa 3 or
m + n & z 7
4r 3 , which is obviously the case.
(ii) Solution of the cubic by Dionysodorus.
It is convenient to add here that Eutocius gives, in addition
to the solution by Archimedes, two other solutions of our
problem. One, by Dionysodorus, solves the cubic equation in
the less general form in which it is required for Archimedes's
proposition. This form, obtained from (8) above, by putting
A'M = x, is
4r 2 :x 2 = (3r— x)
n
m + n
r,
and the solution is obtained by drawing the parabola and
the rectangular hyperbola which we should represent by the
equations
n
r(3r~x) = y 2 and
n
2 r 2 = xy,
m+n m+n
referred to A 'A and the perpendicular to it through A as axes
of x, y respectively.
(We make FA equal to OA, and draw the perpendicular
AH of such a length that
FA:AH = CE:ED = (m + n):n.)
ON THE SPHERE AND CYLINDER, II 47
(iii) Solution of the original problem of II. 4 by Diocles.
Diodes proceeded in a different manner, satisfying, by
a geometrical construction, not the derivative cubic equation,
but the three simultaneous relations which hold in Archi-
medes's proposition, namely
HM:H'M = m:n
HA: h = r :h'
H'A'\ I! = r:hj
with the slight generalization that he substitutes for r in
these equations another length a.
The problem is, given a straight line A A', sl ratio m : n, and
another straight line A K (= a), to divide A A' at a point M
and at the same time to find two points H, ffl on A A'
produced such that the above relations (with a in place
of r) hold.
The analysis leading to the construction is very ingenious.
Place AK (= a) at right angles to AA', and draw A'K' equal
and parallel to it.
Suppose the problem solved, and the points M, H, H f all
found.
Join KM, produce it, and complete the rectangle KGEK f .
48 ARCHIMEDES
Draw QMN through M parallel to AK. Produce K'M to
meet KG produced in F.
By similar triangles,
FA:AM = K'A':A'M, or FA:h = a:h' y
whence FA = AH (k, suppose).
Similarly # A'E — A'H' (Jc, suppose).
Again, by ^similar triangles,
(FA + AM) : (A'K' + A'M) = AM: A'M
= (AK + AM):(EA' + A'M),
or (k + h):(a + h') = (a + h):(¥ + h') }
i. e. (k + h) (k' + h') = (a + h)(a + h f ). (1)
Now, by hypothesis,
m:n = (k + h):(k' + hf)
= (k + h)(k' + h'):(k' + h') 2
= (a + h) (a + hf) : {¥ + h!f [by (I)]. (2)
Measure AR, A f R! on AA f produced both ways equal to a.
Draw RP, R'P' at right angles to RR / as shown in the figure.
Measure along MJS T the length M V equal to MA' or h\ and
draw PP' through V, A' to meet RP, R'P'.
Then QV=k' + h', P'V= V2 (a + h'),
PV= V2(a + h),
whence PV.P'V = 2 (a + h) {a + h!) ;
and, from (2) above,
2m:n=2(a + h) (a + h') : (¥ + h'f
= PV.P'V: QV 2 . (3)
Therefore Q is on an ellipse in which PP' is a diameter, and
Q V is an ordinate to it.
Again, □ GQNK is equal to □ AA'K'K, whence
GQ.QN= AA'. A'K' =(h + h!) a = 2ra, (4)
and therefore Q is on the rectangular hyperbola with KF,
KK' as asymptotes and passing through A'.
ON THE SPHERE AND CYLINDER, II 49
How this ingenious analysis was suggested it is not possible
to say. It is the equivalent of reducing the four unknowns
h, hf, k, k' to two, by putting h = r + x, h' = r—x and h' = y,
and then reducing the given relations to two equations in x, y,
which are coordinates of a point in relation to Ox, Oy as axes,
where is the middle point of AA\ and Ox lies along 0A\
while Oy is perpendicular to it.
Our original relations (p. 47) give
-., ah' r — x 7 ah r + x , m h + k
y = fc = —r- = a j k = 77 = a j and — = t-. — ^ •
h r + x h r — x n h +k
We have at once, from the first two equations,
Icy = a y = a ,
whence {r + x)y = a (r — x),
and (x + r) (y + a) = 2 ra,
which is the rectangular hyperbola (4) above.
, ; (r + x)(l + -^—)
. m h + k '\ r — x/
6 ' n "" W + kf
(r-x)(l+^-)
r + x,
whence we obtain a cubic equation in x,
(r + x) 2 {r + a — x) = — (r — &) 2 (r + c& + x),
which gives
m / w/^+a + arv 2
— (r— o;W
w v V r +
(r— x) 2 (~ -I =(r + a) 2 — x 2 .
t, , 1/ a . y + r — x r + a + x
But ~^— = , whence = >
r — & r + # r — a; r + #;
and the equation becomes
— (y + r — x) 2 = (r + a) 2 — x 2 ,
which is the ellipse (3) above.
1523.2 E
50 ARCHIMEDES
To return to Archimedes. Book II of our treatise contains
further problems : To find a sphere equal to a given cone or
cylinder (Prop. 1), solved by reduction to the finding of two
mean proportionals; to cut a sphere by a plane into two
segments having their surfaces in a given ratio (Prop. 3),
which is easy (by means of I. 42, 43) ; given two segments of
spheres, to find a third segment of a sphere similar to one
of the given segments and having its surface equal to that of
the other (Prop. 6) ; the same problem with volume substituted
for surface (Prop. 5), which is again reduced to the finding
of two mean proportionals; from a given sphere to* cut off
a segment having a given ratio to the cone with the same
base and equal height (Prop. 7). The Book concludes with
two interesting theorems. If a sphere be cut by a plane into
two segments, the greater of which has its surface equal to S
and its volume equal to V, while S', Y f are the surface and
volume of the lesser, then V: V < S 2 : S' 2 but > S*:S'i
(Prop. 8) : and, of all segments of spheres which have their
surfaces equal, the hemisphere is the greatest in volume
(Prop. 9).
Measurement of a Circle.
The book on the Measurement of a Circle consists of three
propositions only, and is not in its original form, having lost
(as the treatise On the Sphere and Cylinder also has) prac-
tically all trace of the Doric dialect in which Archimedes
wrote ; it may be only a fragment of a larger treatise. The
three propositions which survive prove (1) that the area of
a circle is equal to that of a right-angled triangle in which
the perpendicular is equal to the radius, and the base to the
circumference, of the circle, (2) that the area of a circle is to
the square on its diameter as 11 to 14 (the text of this pro-
position is, however, unsatisfactory, and it cannot have been
placed by Archimedes before Prop. 3, on which it depends),
(3) that the ratio of the circumference of any circle to its
diameter (i.e. n) is < 3y but > 3-^f. Prop. 1 is proved by
the method of exhaustion in Archimedes's usual form : he
approximates to the area of the circle in both directions
(a) by inscribing successive regular polygons with a number of
MEASUREMENT OF A CIRCLE 51
sides continually doubled, beginning from a square, (b) by
circumscribing a similar set of regular polygons beginning
from a square, it being shown that, if the number of the
sides of these polygons be continually doubled, more than half
of the portion of the polygon outside the circle will be taken
away each time, so that we shall ultimately arrive at a circum-
scribed polygon greater than the circle by a space less than
any assigned area.
Prop. 3, containing the arithmetical approximation to n, is
the most interesting. The method amounts to calculating
approximately the perimeter of two regular polygons of 96
sides, one of which is circumscribed, and the other inscribed,
to the circle ; and the calculation starts from a greater and
a lesser limit to the value of V 3, which Archimedes assumes
without remark as known, namely
265 s- a/Q ^ 1351
IT'S <~ V O <. -780-.
How did Archimedes arrive at these particular approxi-
mations? No puzzle has exercised more fascination upon
writers interested in the history of mathematics. De Lagny,
Mollweide, Buzengeiger, Hauber, Zeuthen, P. Tannery, Heiler-
mann, Hultsch, Hunrath, Wertheim, Bobynin : these are the
names of some of the authors of different conjectures. The
simplest supposition is certainly that of Hunrath and Hultsch,
who suggested that the formula used was
b h
a ± ~- > V(a?± b) > a ±
2a v -*■ ' — 2a+ 1
where a 2 is the nearest square number above or below a 2 ± b,
as the case may be. The use of the first part of this formula
by Heron, who made a number of such approximations, is
proved by a passage in his Metrica 1 , where a rule equivalent
to this is applied to \/720 ; the second part of the formula is
used by the Arabian Alkarkhi (eleventh century) who drew
from Greek sources, and one approximation in Heron may be
obtained in this way. 2 Another suggestion (that of Tannery
1 Heron, Metrica, i. 8.
2 Stereom. ii, p. 184. 19, Hultsch; p. 154. 19, Heib. ^54 = 7^ = 7^
instead of 7 I 5 1 .
E 2
52 ARCHIMEDES
and Zeuthen) is that the successive solutions in integers of
the equations
x 2 — 3 y 2 = 1
x 2 -3y<
-Sy 2 = 1 i
-3y 2 = -2)
may have been found in a similar way to those of the
equations x 2 — 2y 2 = ±1 given by Theon of Smyrna after
the Pythagoreans. The rest of the suggestions amount for the
most part to the use of the method of continued fractions
more or less disguised.
Applying the above formula, we easily find
2-i> V3 >2~§,
or | > \/3 > §.
Next, clearing of fractions, we consider 5 as an approxi-
mation to V 3 . 3 2 or a/27, and we have
5 + T 2 o > 3 ^3 > 6 + x a T ,
whence f| > V 3 > yy.
Clearing of fractions again, and taking 26 as an approxi-
mation to \/3 A5 2 or \/675, we have
26— & > 15^3 > 26-sV,
which reduces to
135
78
;1 ^ J*\ ~> 265
q- > V 6 > y-g-j.
Archimedes first takes the case of the circumscribed polygon.
Let CA be the tangent at J. to a circular arc with centre 0.
Make the angle AOG equal to one-third of a right angle.
Bisect the angle AOG by OD, the angle AOD by OE, the
angle AOE by OF, and the angle AOFby OG. Produce GA
to AH, making AH equal to AG. The angle GOH is then
equal to the angle FOA which is ^th of a right angle, so
that GH is the side of a circumscribed regular polygon with
96 sides.
Now OA :AG[ = \/3:l] > 265:153, (1)
and OG : C A = 2:1 = 306:153. (2)
MEASUREMENT OF A CIRCLE
And, since OD bisects the angle CO A,
CO:OA = CD: DA,
so that (CO + OA):OA = CA: DA,
or (CO + OA) :CA = OA: AD.
Hence OA : AD > 571 : 153,
53
by (1) and (2).
And OD 2 : AD 2 = (OA 2 + AD 2 ) : AD 2
> (571 2 +153 2 ): 153 2
> 349450: 23409.
Therefore, says Archimedes,
OD.DA > 591$: 153.
Next, just as we have found the limit of OD : AD
from OC : CA and the limit of OA : AC, we find the limits
of OA-.AE and OE-.AE from the limits of 0D:DA and
Oil : ^.D, and so on. This gives ultimately the limit of
0A:AG.
Dealing with the inscribed polygon, Archimedes gets a
similar series of approximations. ABC being a semicircle, the
angle BAC is made equal to one-third of a right angle. Then,
if the angle BAC is bisected by AD, the angle BAD by AE,
the angle BAE by AF, and the angle BAF by AG, the
straight line BG is the side of an inscribed polygon with
96 sides. r
54 ARCHIMEDES
Now the triangles ADB, BDd, ACd are similar;
therefore AD : DB = BD : Dd = AC : Cd
= AB: Bd, since AD bisects Z BAG,
= (AB + AC):(Bd + Cd)
= (AB + AC):BC.
But AG : GB < 1351 : 780,
while .RA : jBO = 2 : 1 = 1560 : 780.
Therefore AD : DB < 2911 : 780.
Hence
AB 2 :BD 2 < (2911 2 + 780 2 ):780 2
< 9082321 : 608400,
and, says Archimedes,
AB.BD < 3013|: 780.
Next, just as a limit is found for AD : DB and AB : BD
from AB : BG and the limit of AG: GB, so we find limits for
AE.EB and AB : BE from the limits of AB : BD and AD : Di?,
and so on, and finally we obtain the limit of AB : BG.
We have therefore in both cases two series of terms a , a lf
(t 2 ... a n and b , b l} b 2 ... b n , for which the rule of formation is
a x = a + b , a 2 = a x + &!,... ,
where 6 2 = \/ («/ + c 2 ), 6 2 = \/(a 2 2 + c 2 ) . . . ;
and in the first case
<% = 265, 6 = 306, c=153,
while in the second case
a t
1351, b n - 1560, c = 780.
MEASUREMENT OF A CIRCLE
55
The series of values found by Archimedes are shown in the
following table :
a be
265 306 153
571 > |V(571 2 +153 2 )] 153
> 591|
1162|>[V{(1162|) 2 +153 2 }] 153 2
>1172|
2334i>[v / {(2334£) 2 +153 2 }] 153 3
> 2339J
n a b c
1351 1560 780
1 2911 < </(2911 2 + 780 2 ) 780
< 3013JJ
f5924j|
1823 (
4673|
153 4
780]*
< v / (1823 2 +240 2 ) 240/
1 < 1838 T 9 T
3661 T 9 T ... 240 f
1007 (< \/(1007 2 + 66 2 ) 66
( < 1009J
< V{(2016^) 2 + 66 2 } 66
2016|
< 2017f
and, bearing in mind that in the first case the final ratio
a 4 : c is the ratio A : AG = 2 OA : OH, and in the second case
the final ratio 6 4 : c is the ratio AB : BG, while GH in the first
figure and BG in the second are the sides of regular polygons
of 96 sides circumscribed and inscribed respectively, we have
finally
96X153 96x66
> 7T >
4673|
2017J
Archimedes simply infers from this that
3i >tt > 3if .
As a matter of fact
96 x 153
4673J
667|
667 * _ i
= 3 4673|' and 4672J "
1
It is also to be observed that 3^£ = 3 -\ -, and it may
have been arrived at by a method equivalent to developing
the fraction
6336
2017j
in the form of a continued fraction.
It should be noted that, in the text as we have it, the values
of b lf b 2 , 63, 6 4 are simply stated in their final form without
the intermediate step containing the radical except in the first
* t Here the ratios of a to c are in the first instance reduced to lower
terms.
56 ARCHIMEDES
case of all, where we are told that 0D l :AD 2 > 349450 : 23409
and then that OD.DA > 591j:153. At the points marked
* and f in the table Archimedes simplifies the ratio a 2 : c and
a z : c before calculating b 2 , b z respectively, by multiplying each
term in the first case by 5 % and in the second case by JJ.
He gives no explanation of the exact figure taken as the
approximation to the square root in each case, or of the
method by which he obtained it. We may, however, be sure
that the method amounted to the use of the formula (a±b) 2
= a 2 + 2 ab + b 2 , much as our method of extracting the square
root also depends upon it.
We have already seen (vol. i, p. 232) that, according to
Heron, Archimedes made a still closer approximation to the
value of 77.
On Conoids and Spheroids.
The main problems attacked in this treatise are, in Archi-
medes's manner, stated in his preface addressed to Dositheus,
which also sets out the premisses with regard to the solid
figures in question. These premisses consist of definitions and
obvious inferences from them. The figures are (1) the right-
angled conoid (paraboloid of revolution), (2) the obtuse-angled
conoid (hyperboloid of revolution), and (3) the spheroids
(a) the oblong, described by the revolution of an ellipse about
its 'greater diameter' (major axis), (b) the flat, described by
the revolution of an ellipse about its ' lesser diameter ' (minor
axis). Other definitions are those of the vertex and axis of the
figures or segments thereof, the vertex of a segment being
the point of contact of the tangent plane to the solid which
is parallel to the base of the segment. The centre is only
recognized in the case of the spheroid ; what corresponds to
the centre in the case of the hyperboloid is the ' vertex of
the enveloping cone ' (described by the revolution of what
Archimedes calls the 'nearest lines to the section of the
obtuse-angled cone', i.e. the asymptotes of the hyperbola),
and the line between this point and the vertex of the hyper-
boloid or segment is called, not the axis or diameter, but (the
line) 'adjacent to the axis'. The axis of the segment is in
the case of the paraboloid the line through the vertex of the
segment parallel to the axis of the paraboloid, in the case
ON CONOIDS AND SPHEROIDS 57
of the hyperboloid the portion within the solid of the line
joining the vertex of the enveloping cone to the vertex of
the segment and produced, and in the case of the spheroids the
line joining the points of contact of the two tangent planes
parallel to the base of the segment. Definitions are added of
a ' segment of a cone ' (the figure cut off towards the vertex by
an elliptical, not circular, section of the cone) and a ' frustum
of a cylinder' (cut off by two parallel elliptical sections).
Props. 1 to 1 8 with a Lemma at the beginning are preliminary
to the main subject of the treatise. The Lemma and Props. 1, 2
are general propositions needed afterwards. They include
propositions in summation,
2 {a + 2a + 3a+ ... + na} > n.na > 2{a + 2a+ ... + (n—l)a}
(Lemma)
(this is clear from S n = ^n(n + I) a) ;
(n + 1) (na) 2 +-a(a + 2a + 3a + ... + na)
= 3 {a 2 + (2a) 2 + (3a) 2 + ... + (no) 2 } ;
(Lemma to Prop. 2)
whence (Cor.)
3 {a 2 + (2a) 2 + (3a) 2 + ... +(na) 2 } > n(na) 2
> 3{a 2 + (2a) 2 + ... + (n-la) 2 } ;
lastly, Prop. 2 gives limits for the sum of n terms of the
series ax + x 2 , a.2x + (2x) 2 , a . 3 x + (3 x) 2 , . . . , in the form of
inequalities of ratios, thus :
n{a.nx + (nx) 2 } : 2/'" 1 {a.rx + (rx) 2 }
> (a + nx) : (\a + \nx)
> n { a . nx + (nx) 2 } : 2 X W { a . rx + (rx) 2 } .
Prop. 3 proves that, if QQ' be a chord of a parabola bisected
at Fby the diameter PV, then, if PV be of constant length,
the areas of the triangle PQQ' and of the segment PQQ f are
also constant, whatever be the direction of QQ' ; to prove it
Archimedes assumes a proposition ' proved in the conies ' and
by no means easy, namely that, if QD be perpendicular to PV,
and if p, p a be the parameters corresponding to the ordinates
parallel to QQ' and the principal ordinates respectively, then
Props. 4-6 deal with the area of an ellipse, which is, in the
58 ARCHIMEDES
first of the three propositions, proved to be to the area of
the auxiliary circle as the minor axis to the major ; equilateral
polygons of 4 n sides are inscribed in the circle and compared
with corresponding polygons inscribed in the ellipse, which are
determined by the intersections with the ellipse of the double
ordinates passing through the angular points of the polygons
inscribed in the circle, and the method of exhaustion is then
applied in the usual way. Props. 7, 8 show how, given an ellipse
with centre C and a straight line CO in a plane perpendicular to
that of the ellipse and passing through an axis of it, (1) in the
case where OC is perpendicular to that axis, (2) in the case
where it is not, we can find an (in general oblique) circular
cone with vertex such that the given ellipse is a section of it,
or, in other words, how we can find the circular sections of the
cone with vertex which passes through the circumference of
the ellipse ; similarly Prop. 9 shows how to find the circular
sections of a cylinder with CO as axis and with surface passing
through the circumference of an ellipse with centre C, where
CO is in the plane through an axis of the ellipse and perpen-
dicular to its plane, but is not itself perpendicular to that
axis. Props. 11-18 give simple properties of the conoids and
spheroids, easily derivable from the properties of the respective
conies; they explain the nature and relation of the sections
made by planes cutting the solids respectively in different ways
(planes through the axis, parallel to the axis, through the centre
or the vertex of the enveloping cone, perpendicular to the axis,
or cutting it obliquely, respectively), with especial reference to
the elliptical sections of each solid, the similarity of parallel
elliptical sections, &c. Then with Prop. 19 the real business
of the treatise begins, namely the investigation of the volume
of segments (right or oblique) of the two conoids and the
spheroids respectively.
The method is, in all cases, to circumscribe and inscribe to
the segment solid figures made up of cylinders or ' frusta of
cylinders ', which can be made to differ as little as we please
from one another, so that the circumscribed and inscribed
figures are, as it were, compressed together and into coincidence
with the segment which is intermediate between them.
In each diagram the plane of the paper is a plane through
the axis of the conoid or spheroid at right angles to the plane
ON CONOIDS AND SPHEROIDS
59
of the section which is the base of the segment, and which
is a circle or an ellipse according as the said base is or is not
at right angles to the axis ; the plane of the paper cuts the
base in a diameter of the circle or an axis of the ellipse as
the case may be.
The nature of the inscribed and circumscribed figures will
be seen from the above figures showing segments of a para-
boloid, a hyperboloid and a spheroid respectively, cut off
60 ARCHIMEDES
by planes obliquely inclined to the axis. The base of the
segment is an ellipse in which BB' is an axis, and its plane is
at right angles to the plane of the paper, which passes through
the axis of the solid and cuts it in a parabola, a hyperbola, or
an ellipse respectively. The axis of the segment is cut into a
number of equal parts in each case, and planes are drawn
through each point of section parallel to the base, cutting the
solid in ellipses, similar to the base, in which PP', QQ', &c, are
axes. Describing frusta of cylinders with axis AD and passing
through these elliptical sections respectively, we draw the
circumscribed and inscribed solids consisting of these frusta.
It is evident that, beginning from A, the first inscribed frustum
is equal to the first circumscribed frustum, the second to the
second, and so on, but there is one more circumscribed frustum
than inscribed, and the difference between the circumscribed
and inscribed solids is equal to the last frustum of which BB'
is the base, and ND is the axis. Since ND can be made as
small as we please, the difference between the circumscribed
and inscribed solids can be made less than any assigned solid
whatever. Hence we have the requirements for applying the
method of exhaustion.
Consider now separately the cases of the paraboloid, the
hyperboloid and the spheroid.
I. The 'paraboloid (Props. 20-22).
The frustum the base of which is the ellipse in which PP' is
an axis is proportional to PP' 2 or PN 2 , i.e. proportional to
AX. Suppose that the axis AD (= c) is divided into n equal
parts. Archimedes compares each frustum in the inscribed
and circumscribed figure with the frustum of the whole cylinder
BF cut oft' by the same planes. Thus
(first frustum in BF) : (first frustum in inscribed figure)
= BD 2 : PN 2
= AD:AN
= BD : TK
Similarly
(second frustum in BF) : (second in inscribed figure)
= HN:3M,
and so on. The last frustum in the cylinder BF has none to
ON CONOIDS AND SPHEROIDS 61
correspond to it in the inscribed figure, and we should write
the ratio as (BD : zero).
Archimedes concludes, by means of a lemma in proportions
forming Prop. 1, that
(frustum BF) : (inscribed figure)
= (BD + HN+ ...) :(TN + SM+ ...+ XO)
= n 2 k : (k + 2k + 3k + ... + n-lk),
where XO = k, so that BD = nk.
In like manner, he concludes that
(frustum BF) : (circumscribed figure)
= M 2 Jc : (Jc 4- 2 h + 3 k + . . . + ?i&).
But, by the Lemma preceding Prop. 1,
/c + 2/s+3&+...+w— Ik < ^n z k < k+2k+3k+ ... +w&,
whence
(frustum i?jP) : (inscr. fig.) > 2 > (frustum BF) : (circumscr. fig.).
This indicates the desired result, which is then confirmed by
the method of exhaustion, namely that
(frustum BF) = 2 (segment of paraboloid),
or, if V be the volume of the ' segment of a cone ', with vertex
A and base the same as that of the segment,
(volume of segment) = ^V.
Archimedes, it will be seen, proves in effect that, if k be
indefinitely diminished, and n indefinitely increased, while nk
remains equal to c, then
limit of k{k+2k + 3k+ ... +(n— l)k} = \6\
that is, in our notation,
JU(a/JU — ■?> •
Jo
Prop. 23 proves that the volume is constant for a given
length of axis AD, whether the segment is cut off* by a plane
perpendicular or not perpendicular to the axis, and Prop. 24
shows that the volumes of two segments are as the squares on
their axes.
62 ARCHIMEDES
II. In the case of the hyperboloid (Props. 25, 26) let the axis
AD be divided into n parts, each of length h, and let AA'=a.
Then the ratio of the volume of the frustum of a cylinder on
the ellipse of which any double ordinate QQ' is an axis to the
volume of the corresponding portion of the whole frustum BF
takes a different form ; for, if AM = rh, we have
(frustum in BF) : (frustum on base QQ')
= BD 2 : QM 2
= AD . A'D : AM . A'M
— [a.nh + (nh) 2 } : {a . rh+ (rh) 2 }.
By means of this relation Archimedes proves that
(frustum BF) : (inscribed figure)
= n {a.nh+ (nh) 2 } : S^- 1 { a . rh + (rh) 2 } ,
and
(frustum BF) : (circumscribed figure)
= n{a.nh+(nh) 2 } : 2 x n {a.rh-\-(rh) 2 }.
But, by Prop. 2,
n{a.nh + (nh) 2 } :\ n ~ l {a.rh + (rh) 2 } > (a + nh):(±a + %nh)
> n{a.nh + (nh) 2 } :2 { n {a.rh-\-(rh) 2 }.
From these relations it is inferred that
(frustum BF) : (volume of segment) = (a + nh) : (^a + ^nh),
or (volume of segment) : (volume of cone ABB')
= (AD+3CA):(AD + 2CA);
and this is confirmed by the method of exhaustion.
The result obtained by Archimedes is equivalent to proving
that, if h be indefinitely diminished while n- is indefinitely
increased but nh remains always equal to b, then
limit of n(ab + b 2 )/S n = (a + b) / '(£« + §&),
or limit of - S n = b 2 (\a + J b),
it
where
S n = a(h + 2h+3h+...+nh) + {h 2 + (2h) 2 + (3h) 2 +...+(nh) 2 }
ON CONOIDS AND SPHEROIDS 63
so that
hS n = ah(h + 2h+...+nh) + h{h 2 + (2h) 2 +...+(nh) 2 }.
The limit of this latter expression is what we should write
nb
(ax + x 2 ) dx = b 2 (%a + §6),
Jo
and Archimedes's procedure is the equivalent of this integration.
III. In the case of the spheroid (Props. 29, 30) we take
a segment less than half the spheroid.
As in the case of the hyper boloid,
(frustum in BF) : (frustum on base QQ')
= BD 2 : QM 2
= AD.A'D:AM.A'M;
but, in order to reduce the summation to the same as that in
Prop. 2, Archimedes expresses AM . A'M in a different form
equivalent to the following.
Let AD (=b) be divided into n equal parts of length h,
and suppose that AA f — a, CD = \c.
Then AD.A'D = ±a 2 -±c 2 ,
and AM . A'M = \a 2 - (Jc + rhf {DM = rh)
= AD . A'D-{c . rh + (rh) 2 }
= cb + b 2 -{c.rh + (rh) 2 }.
Thus in this case we have
(frustum BF) : (inscribed figure)
= n(cb + b 2 ) : [n(cb + b 2 ) - 2^{c . rh + (rh) 2 }]
and
(frustum BF) : (circumscribed figure)
= n (cb + b 2 ) : [n (cb + b 2 ) - S^" 1 {c.rh + (rh) 2 } ].
And, since b = nh, we have, by means of Prop. 2,
n(cb + b 2 ) : [n(cb + b 2 ) -^{c. rh + (rh) 2 }]
>(c + 6):{c + 6-(i C + |6)}
> n(cb + b 2 ) : [n(cb + b 2 ) ^^ l n ~ 1 {c . rh + (rh) 2 }].
64 ARCHIMEDES
The conclusion, confirmed as usual by the method of ex-
haustion, is that
(frustum BF) : (segment of spheroid) = (c + b) : { c + b - (%v + J b) }
= (c + 6):(|c + #6) 5
whence (volume of segment) : (volume of cone ABB')
=:(|c + 26):(c + 6)
= (3GA-AD):(2GA-AD), since CA = ±c + b.
As a particular case (Props. 27, 28), half the spheroid is
double of the corresponding cone.
Props. 31, 32, concluding the treatise, deduce the similar
formula for the volume of the greater segment, namely, in our
figure,
(greater segmt.) : (cone or segmt.of cone with same base and axis)
= (CA + AD):AD.
On Spirals.
The treatise On Spirals begins with a preface addressed to
Dositheus in which Archimedes mentions the death of Conon
as a grievous loss to mathematics, and then summarizes the
main results of the treatises On the Sphere and Cylinder and
On Conoids and Spheroids, observing that the last two pro-
positions of Book II of the former treatise took the place
of two which, as originally enunciated to Dositheus, were
wrong; lastly, he states the main results of the treatise
On Spirals, premising the definition of a spiral which is as
follows:
' If a straight line one extremity of which remains fixed be
made to revolve at a uniform rate in a plane until it returns
to the position from which it started, and if, at the same time
as the straight line is revolving, a point move at a uniform
rate along the straight line, starting from the fixed extremity,
the point will describe a spiral in the plane.'
As usual, we have a series of propositions preliminary to
the main subject, first two propositions about uniform motion,
ON SPIRALS
65
then two simple geometrical propositions, followed by pro-
positions (5-9) which are all of one type. Prop. 5 states that,
given a circle with centre 0, a tangent to it at A, and c, the
Fig. 1.
circumference of any circle whatever, it is possible to draw
a straight line OPF meeting the circle in P and the tangent
in F such that
FP:OP < (arc AP) : c.
Archimedes takes D a straight line greater than c, draws
OH parallel to the tangent at A and then says ' let PH be
placed equal to D verging (vevovcra) towards A '. This is the
usual phraseology of the type of problem known as vevcris
where a straight line of given length has to be placed between
two lines or curves in such a position that, if produced, it
passes through a given point (this is the meaning of verging) .
Each of the propositions 5-9 depends on a vevcris of this kind,
Fig. 2.
which Archimedes assumes as * possible ' without showing how
it is effected. Except in the case of Prop. 5, the theoretical
solution cannot be effected by means of the straight line and
circle ; it depends in general on the solution of an equation
of the fourth degree, which can be solved by means of the
1523.2 F
66
ARCHIMEDES
points of intersection of a certain rectangular hyperbola
and a certain parabola. It is quite possible, however, that
such problems were in practice often solved by a mechanical
method, namely by placing a ruler, by trial, in the position of
the required line : for it is only necessary to place the ruler
so that it passes through the given point and then turn it
round that point as a pivot till the intercept becomes of the
given length. In Props. 6-9 we have a circle with centre 0,
a chord AB less than the diameter in it, OM the perpendicular
from on AB, BT the tangent at B, OT the straight line
through parallel to AB ; B : E is any ratio less or greater,
as the case may be, than the ratio BM : MO. Props. 6, 7
(Fig. 2) show that it is possible to draw a straight line OFP
Fig. 3.
meeting AB in F and the circle in P such that FP : PB=D : E
(OP meeting AB in the case where D:E < BM:M0, and
meeting AB produced when D : E > BM : MO). In Props. 8, 9
(Fig. 3) it is proved that it is possible to draw a straight line
OFP meeting AB in F, the circle in P and the tangent at B in
G, such that FP:BG = D:E (OP meeting AB itself in the case
where D : E < BM: MO, and meeting AB produced in the
case where D:E > BM : MO).
We will illustrate by the constructions in Props. 7, 8,
as it is these propositions which are actually cited later.
Prop. 7. If D : E is any ratio > BM : MO, it is required (Fig. 2)
to draw 0P / F / meeting the circle in P f and AB produced in
F' so that
F / P / :P / B = D:E.
Draw OT parallel to AB, and let the tangent to the circle at
B meet OT in T.
ON SPIRALS 67
Then D : E > BM : MO, by hypothesis,
> OB : BT, by similar triangles.
Take a straight line P'H' (less than BT) such that D : £
= 05 : P'H', and place P'fT' between the circle and OT
1 verging towards B ' (construction assumed).
Then F'P' : P'B = OP' : P'H'
= OB : P'H'
= D:E.
Prop. 8. If D : E is any given ratio < BM: MO, it is required
to draw OFPG meeting AB in F, the circle in P, and the
tangent at B to the circle in G so that
FP : BG = D : E.
If OT is parallel to AB and meets the tangent at B in 7 1 ,
#¥: MO = OB : BT, by similar triangles,
whence D:E<0B:BT
Produce TB to C, making BO of such length that
D : E = OB : BC,
so that BO > BT.
Describe a circle through the three points 0, T, C and let OB
produced meet this circle in K.
' Then, since BC > BT, and OK is perpendicular to OT, it is
possible to place QG [between the circle TKC and BO] equal to
BK and verging towards ' (construction assumed).
68 ARCHIMEDES
Let QGO meet the original circle in P and AB in F. Then
OFPG is the straight line required.
For CG.GT=OG.GQ = OG. BK.
But OF:OG = BT: GT, by parallels,
whence OF.GT=OG.BT.
Therefore CG . GT : OF . GT = OG . BK : OG . BT,
whence CG:OF= BK : BT
= BG:OB
= P<7 : OP.
Therefore OP:OF=BC: CG,
and Hence PF:OP = BG: BC,
or PP: BG=OB:BC=D: E.
Pappus objects to Archimedes's use of the vevcris assumed in
Prop. 8, 9 in these words :
' it seems to be a grave error into which geometers fall
whenever any one discovers the solution of a plane problem
by means of conies or linear (higher) curves, or generally
solves it by means of a foreign kind, as is the case e.g. (1) with
the problem in the fifth Book of the Conies of Apollonius
relating to the parabola, and (2) when Archimedes assumes in
his work on the spiral a vedais of a "solid" character with
reference to a circle ; for it is possible without calling in the
aid of anything solid to find the proof of the theorem given by
Archimedes, that is, to prove that the circumference of the
circle arrived at in the first revolution is equal to the straight
line drawn at right angles to the initial line to meet the tangent
to the spiral (i.e. the subtangent).'
There is, however, this excuse for Archimedes, that he only
assumes that the problem can be solved and does not assume
the actual solution. Pappus 1 himself gives a solution of the
particular vevcris by means of conies. Apollonius wrote two
Books of vevo-61?, and it is quite possible that by Archimedes's
time there may already have been a collection of such problems
to which tacit reference was permissible.
Prop. 10 repeats the result of the Lemma to Prop. 2 of On
1 Pappus, iv, pp. 298-302.
ON SPIRALS 69
Conoids and Spheroids involving the summation of the series
1 2 + 2 2 -f 3 2 + ... + 7t 2 . Prop 11 proves another proposition in
summation, namely that
(n- 1) (na) 2 : {a 2 + (2 a) 2 + (3 a) 2 + ... + (n- l)a) 2 )
> {naf : {na . a + -§■ (na — a) 2 }
> (n-l)(na) 2 : {(2a) 2 +(3a) 2 -f ... +(na)' 1 ).
The same proposition is also true if the terms of the series
are a 2 , (a + b) 2 , (a + 2b) 2 ... (a + n— lb) 2 , and it is assumed in
the more general form in Props. 25, 26.
Archimedes now introduces his Definitions, of the spiral
itself, the origin, the initial line, the first distance (= the
radius vector at the end of one revolution), the second distance
(= the equal length added to the radius vector during the
second complete revolution), and so on ; the first area (the area
bounded by the spiral described in the first revolution and
the ' first distance '), the second area (that bounded by the spiral
described in the second revolution and the ' second distance '),
and so on; the^irs^ circle (the circle with the 'first distance'
as radius), the second circle (the circle with radius equal to the
sum of the 'first' and 'second distances', or twice the first
distance), and so on.
Props. 12, 14, 15 give the fundamental property of the
spiral connecting the length of the radius vector with the angle
through which the initial line has revolved from its original
position, and corresponding to the equation in polar coordinates
r = a 0. As Archimedes does not speak of angles greater
than 7r, or 27r, he has, in the case of points on any turn after
the first, to use multiples of the circumference
of a circle as well as arcs of it. He uses the
'first circle' for this purpose. Thus, if P, Q
are two points on the first turn,
OP:OQ = (arc AKP') : (arc AKQ') ;
if P, Q are points on the wth turn of the
spiral, and c is the circumference of the first circle,
OP:OQ= {{n-\)c + MeAKP'} : {(n- l)c + arc AKQ'}.
Prop. 1 3 proves that, if a straight line touches the spiral, it
70 ARCHIMEDES
touches it at one point only. For, if possible, let the tangent
at P touch the spiral at another point Q. Then, if we bisect
the angle POQ by OL meeting PQ in L and the spiral in R,
0P + 0Q=20R by the property of the spiral. But by
the property of the triangle (assumed, but easily proved)
OP+OQ > 2 0L, so that OL < OR, and some point of PQ
lies within the spiral. Hence PQ cuts the spiral, which is
contrary to the hypothesis.
Props. 16, 17 prove that the angle made by the tangent
at a point with the radius vector to that point is obtuse on the
' forward ' side, and acute on the ' backward ' side, of the radius
vector.
Props. 18-20 give the fundamental proposition about the
tangent, that is to say, they give the length of the subtangent
at any point P (the distance between and the point of inter-
section of the tangent with the perpendicular from to OP).
Archimedes always deals first with the first turn and then
with any subsequent turn, and with each complete turn before
parts or points of any particular turn. Thus he deals with
tangents in this order, (1) the tangent at A. the end of the first
turn, (2) the tangent at the end of the second and any subse-
quent turn, (3) the tangent at any intermediate point of the
first or any subsequent turn. We will take as illustrative
the case of the tangent at any intermediate point P of the first
turn (Prop. 20).
If OA be the initial line, P any point on the first turn, PT
the tangent at P and OT perpendicular to OP, then it is to be
proved that, if ASP be the circle through P with centre 0,
meeting PT in S, then
(subtangent OT) = (arc ASP).
I. If possible, let OT be greater than the arc ASP.
Measure off OU such that OU > arc ASP but < OT.
Then the ratio PO : OU is greater than the ratio P0 : OT,
i.e. greater than the ratio of %PS to the perpendicular from
on PS.
Therefore (Prop. 7) we can draw a straight line OQF meeting
TP produced in F, and the circle in Q, such that
FQ:PQ = P0:0U.
ON SPIRALS 71
Let OF meet the spiral in Q f .
Then we have, alternando, since PO = QO,
FQ:QO = PQ:OU
< (arc PQ) : (arc ASP), by hypothesis and a fortiori.
Componendo, FO :Q0 < (arc ASQ) : (arc ASP)
< OQ':OP.
*
But QO = OP ; therefore FO < OQ' ; which is impossible.
Therefore OT is not greater than the arc ASP.
II. Next suppose, if possible, that OT < arc ASP.
Measure OF along OT such that OV is greater than OT but
less than the arc ASP.
Then the ratio PO : OV is less than the ratio PO : OT, i.e.
than the ratio of \PS to the perpendicular from on PS;
therefore it is possible (Prop. 8) to draw a straight line OF'RG
meeting PS, the circle PSA, and the tangent to the circle at P
in F\ R, G respectively, and such that
F'R:GP±=PO:OV.
72 ARCHIMEDES
Let OF'G meet the spiral in R'.
Then, since PO = RO, we have, alter nando,
F'R:RO= GP:OV
> (arc PR) : (arc ASP), a fortiori,
whence F'O : RO < (arc ASR) : (arc ASP)
< OR: OP,
so that F'O < OR'; which is impossible.
Therefore OT is not less than the arc ASP. And it was
proved not greater than the same arc. Therefore
0T= (arc ASP).
As particular cases (separately proved by Archimedes), if
P be the extremity of the first turn and c 2 the circumference
of the first circle, the subtangent = c x ; if P be the extremity
of the second turn and c 2 the circumference of the 'second
circle', the subtangent = 2c 2 ; and generally, if c n be the
circumference of the nth circle (the circle with the radius
vector to the extremity of the nth turn as radius), the sub-
tangent to the tangent at the extremity of the nth turn = nc n .
If P is a point on the nth turn, not the extremity, and the
circle with as centre and OP as radius cuts the initial line
in K, while p is the circumference of the circle, the sub-
tangent to the tangent at P = (w,— l)p + arc iTP (measured
' forward ')}
The remainder of the book (Props. 21-8) is devoted to
finding the areas of portions of the spiral and its several
turns cut off by the initial line or any two radii vectores.
We will illustrate by the general case (Prop. 26). Take
OB, OC, two bounding radii vectores, including an arc BG
of the spiral. With centre and radius OC describe a circle.
Divide the angle BOG into any number of equal parts by
radii of this circle. The spiral meets these radii in points
P, Q ... F, Z such that the radii vectores OB, OP, OQ ... OZ, OC
1 On the whole course of Archimedes's proof of the property of the
subtangent, see note in the Appendix.
ON SPIRALS
73
are in arithmetical progression. Draw arcs of circles with
radii OB, OP, OQ ... as shown; this produces a figure circum-
scribed to the spiral and consisting of the sum of small sectors
of circles, and an inscribed figure of the same kind. As the
first sector in the circumscribed figure is equal to the second
sector in the inscribed, it is easily seen that the areas of the
circumscribed and inscribed figures differ by the difference
between the sectors OzG and OBp' '; therefore, by increasing
the number of divisions of the angle BOO, we can make the
difference between the areas of the circumscribed and in-
scribed figures as small as we please ; we have, therefore, the
elements necessary for the application of the method of
exhaustion.
If there are n radii OB, OP ... 00, there are (n—1) parts of
the angle BOG. Since the angles of all the small sectors are
equal, the sectors are as the square on their radii.
Thus (whole sector Ob' 0) : (circumscribed figure)
= (n- l)OC 2 : (OP 2 + OQ 2 + ... + OC 2 ),
and (whole sector Ob'C) : (inscribed figure)
= (n-l)0C*:(0B 2 + 0P* + 0Q 2 +... + 0Z 2 ).
74, ARCHIMEDES
And OB, OP, OQ, . . . OZ, OG is an arithmetical progression
of n terms; therefore (cf. Prop. 11 and Cor.),
(n- \)0G 2 : (OP 2 + OQ 2 + ... + OG 2 )
< OC 2 :{OC.OB + i(OC-OB) 2 }
< (n-l)OC 2 :(OB 2 + OP 2 +... + OZ 2 ).
Compressing the circumscribed and inscribed figures together
in the usual way, Archimedes proves by exhaustion that
(sector Ob'C) : (area of spiral OBC)
= 00 2 : {OC. OB + ^(00 -OB) 2 }.
If OB = b, OG = c, and (c— b) = (n— l)h, Archimedes's
result is the equivalent of saying that, when h diminishes and
n increases indefinitely, while c — b remains constant,
limit of h{b 2 + (b + h) 2 + (b + 2h) 2 +...+{b + '^2h) 2 }
= {c-b){cb + l;(c-b) 2 }
= §(o 3 -6 3 );
that, is, with our notation,
x 2 dx = l(c 3 — o 3 ).
Jb
In particular, the area included by the first turn and the
initial line is bounded by the radii vectores and 2ira\
the area, therefore, is to the circle with radius 2 it a as ^(2ttcl) 2
to (27ra) 2 , that is to say, it is § of the circle or ^ir(2iTa) 2 .
This is separately proved in Prop. 24 by means of Prop. 10
and Corr. 1, 2.
The area of the ring added while the radius vector describes
the second turn is the area bounded by the radii vectores 2 wet,
and lira, and is to the circle with radius Aw a in the ratio
of {^ 2 r i + 3( r 2~ r i) 2 } t° r 2 , where r x = 2na and r 2 = 47ra;
the ratio is 7:12 (Prop. 25).
If R 1 be the area of the first turn of the spiral bounded by
the initial line, R 2 the area of the ring added by the second
complete turn, R z that of the ring added by the third turn,
and so on, then (Prop. 27)
R 3 == 2R 2) R A = 3R 2 , R ro = 4i? 2 , ... R n = (n-1)R 2 .
Also R 2 = 6R X .
ON SPIRALS 75
Lastly, it' E be the portion of the sector b'OC bounded by
b'B, the arc b'zC of the circle and the arc BC of the spiral, and
F the portion cut off between the arc BC of the spiral, the
radius 00 and the arc intercepted between OB and 00 of
the circle with centre and radius OB, it is proved that
E:F= {0B + %(0C-0B)}:{0B + i(0C-0B)} (Prop. 28).
On Plane Equilibriums, I, II.
In this treatise we have the fundamental principles of
mechanics established by the methods of geometry in its
strictest sense. There were doubtless earlier treatises on
mechanics, but it may be assumed that none of them had
been worked out with such geometrical rigour. Archimedes
begins with seven Postulates including the following prin-
ciples. Equal weights at equal distances balance ; if unequal
weights operate at equal distances, the larger weighs down
the smaller. If when equal weights are in equilibrium some-
thing be added to, or subtracted from, one of them, equilibrium
is not maintained but the weight which is increased or is not
diminished prevails. When equal and similar plane figures
coincide if applied to one another, their centres of gravity
similarly coincide ; and in figures which are unequal but
similar the centres of gravity will be ' similarly situated '.
In any figure the contour of which is concave in one and the
same direction the centre of gravity must be within the figure.
Simple propositions (1-5) follow, deduced by reductio ad
absurdum; these lead to the fundamental theorem, proved
first for commensurable and then by reductio ad absurdum
for incommensurable magnitudes, that Two magnitudes,
whether commensurable or incommensurable, balance at dis-
tances reciprocally proportional to the magnitudes (Props.
6, 7). Prop. 8 shows how to find the centre of gravity of
a part of a magnitude when the centres of gravity of the
other part and of the whole magnitude are given. Archimedes
then addresses himself to the main problems of Book I, namely
to find the centres of gravity of (1) a parallelogram (Props.
9, 10), (2) a triangle (Props. 13, 14), and (3) a parallel-
trapezium (Prop. 15), and here we have an illustration of the
extraordinary rigour which he requires in his geometrical
76 ARCHIMEDES
proofs. We do not find him here assuming, as in The Method,
that, if all the lines that can be drawn in a figure parallel to
(and including) one side have their middle points in a straight
line, the centre of gravity must lie somewhere on that straight
line ; he is not content to regard the figure as made up of an
infinity of such parallel lines ; pure geometry realizes that
the parallelogram is made up of elementary parallelograms,
indefinitely narrow if you please, but still parallelograms, and
the triangle of elementary trapezia, not straight lines, so
that to assume directly that the centre of gravity lies on the
straight line bisecting the parallelograms would really be
a i^etitio principii. Accordingly the result, no doubt dis-
covered in the informal way, is clinched by a proof by reductio
ad absardum in each case. In the case of the parallelogram
ABCD (Prop. 9), if the centre of gravity is not on the straight
line EF bisecting two opposite sides, let it be at H. Draw
HK parallel to AD. Then it is possible by bisecting AE, ED,
then bisecting the halves, and so on, ultimately to reach
a length less than KH. Let this be done, and through the
points of division of AD draw parallels to A B or DC making
a number of equal and similar parallelograms as in the figure.
The centre of gravity of each of these parallelograms is
similarly situated with regard to it. Hence we have a number
of equal magnitudes with their centres of gravity at equal
distances along a straight line. Therefore the centre of
gravity of the whole is on the line joining the centres of gravity
of the two middle parallelograms (Prop. 5, Cor. 2). But this
is impossible, because H is outside those parallelograms.
Therefore the centre of gravity cannot but lie on EF.
Similarly the centre of gravity lies on the straight line
bisecting the other opposite sides AB, CD; therefore it lies at
the intersection of this line with EF, i.e. at the point of
intersection of the diagonals.
ON PLANE EQUILIBRIUMS, I 77
The proof in the case of the triangle is similar (Prop. 13).
Let AD be the median through A. The centre of gravity
must lie on AD.
For, if not, let it be at H, and draw HI parallel to BG.
Then, if we bisect DC, then bisect the halves, and so on,
we shall arrive at a length DE less than IH. Divide BG into
lengths equal to DE, draw parallels to DA through the points
of division, and complete the small parallelograms as shown in
the figure.
The centres of gravity of the whole parallelograms SN, TP,
FQ lie on AD (Prop. 9) ; therefore the centre of gravity of the
figure formed by them all lies on AD; let it be 0. Join OH,
and produce it to meet in Fthe parallel through G to AD.
Now it is easy to see that, if n be the number of parts into
which DG, AC are divided respectively,
(sum of small As AMR, MLS... ARN, NUP ...) : (A ABC)
= n. AN 2 :AC 2
= 1 : n ;
whence
(sum of small As) : (sum of parallelograms) = 1 : (n— 1).
Therefore the centre of gravity of the figure made up of all
the small triangles is at a point X on OH produced such that
XH=(n-l)OH.
But VH: HO <CE: ED or (n - 1) : 1 ; therefore XH > VH.
It follows that the centre of gravity of all the small
triangles taken together lies at X notwithstanding that all
the triangles lie on one side of the parallel to AD drawn
through X : which is impossible.
78 ARCHIMEDES
Hence the centre of gravity of the whole triangle cannot
but lie on AD.
It lies, similarly, on either of the other two medians ; so
that it is at the intersection of any two medians (Prop. 14).
Archimedes gives alternative proofs of a direct character,
both for the parallelogram and the triangle, depending on the
postulate that the centres of gravity of similar figures are
' similarly situated ' in regard to them (Prop. 1 for the
parallelogram, Props. 11, 12 and part 2 of Prop. 13 for the
triangle) .
The geometry of Prop. 15 deducing the centre of gravity of
a trapezium is also interesting. It is proved that, if AD, BG
are the parallel sides (AD being the smaller), and EF is the
straight line joining their middle points, the centre of gravity
is at a point G on EF such that
GE: GF=(2BG + AD): (2AD + BC).
Book II of the treatise is entirely devoted to finding the
centres of gravity of a parabolic segment (Props. 1-8) and
of a portion of it cut off by a parallel to the base (Props. 9, 10).
Prop. 1 (really a particular case of I. 6, 7) proves that, if P, F
^
be the areas of two parabolic segments and D, E their centres
of gravity, the centre of gravity of both taken together is
at a point G on DE such that
P:P'=GE:GD.
ON PLANE EQUILIBRIUMS, I, II 79
This is merely preliminary. Then begins the real argument,
the course of which is characteristic and deserves to be set out.
Archimedes uses a series of figures inscribed to the segment,
as he says, ' in the recognized manner' (yvcopifM<os). The rule
is as follows. Inscribe in the segment the triangle ABB' with
the same base and height; the vertex A is then the point
of contact of the tangent parallel to BB'. Do the same with
the remaining segments cut off by AB, AB', then with the
segments remaining, and so on. If BRQPAP'Q'R'B' is such
a figure, the diameters through Q, Q', P, P', R, R / bisect the
straight lines AB, AB', AQ, AQ', QB, Q'B' respectively, and
BB' is divided by the diameters into parts which are all
equal. It is easy to prove also that PP', QQ', RR' are all
parallel to BB', and that AL : LM: MN: NO = 1 : 3 : 5 : 7, the
same relation holding if the number of sides of the polygon
is increased; i.e. the segments of AO are always in the ratio
of the successive odd numbers (Lemmas to Prop. 2). The
centre of gravity of the inscribed figure lies on AO (Prop. 2).
If there be two parabolic segments, and two figures inscribed
in them ' in the recognized manner ' with an equal number of
sides, the centres of gravity divide the respective axes in the
same proportion, for the ratio depends on the same ratio of odd
numbers 1:3:5:7... (Prop. 3). The centre of gravity of the
parabolic segment itself lies on the diameter AO (this is proved
in Prop. 4 by reductio ad absurdum in exactly the same way
as for the triangle in I. 13). It is next proved (Prop. 5) that
the centre of gravity of the segment is nearer to the vertex A
than the centre of gravity of the inscribed figure is ; but that
it is possible to inscribe in the segment in the recognized
manner a figure such that the distance between the centres of
gravity of the segment and of the inscribed figure is less than
any assigned length, for we have only to increase the number
of sides sufficiently (Prop. 6). Incidentally, it is observed in
Prop. 4 that, if in any segment the triangle with the same
base and equal height is inscribed, the triangle is greater than
half the segment, whence it follows that, each time we increase
the number of sides in the inscribed figure, we take away
more than half of the segments remaining over ; and in Prop. 5
that corresponding segments on opposite sides of the axis, e. g.
QRB, Q'R'B' have their axes equal and therefore are equal in
80 ARCHIMEDES
area. Lastly (Prop. 7), if there be two parabolic segments,
their centres of gravity divide their diameters in the same
ratio (Archimedes enunciates this of similar segments only,
but it is true of any two segments and is required of any two
segments in Prop. 8). Prop. 8 now finds the centre of gravity
of any segment by using the last proposition. It is the
geometrical equivalent of the solution of a simple equation in
the ratio (m, say) of AG to A0, where G is the centre of
gravity of the segment.
Since the segment = § (A ABB'), the sum of the two seg-
ments AQB, AQ'B' = ±(AABB').
Further, if QD, Q'D' are the diameters of these segments,
QD, Q'D' are equal, and, since the centres
of gravity H, H' of the segments divide
QD, Q'D' proportionally, HH' is parallel
to QQ', and the centre of gravity of the
two segments together is at K, the point
where HH' meets A0.
Now A0 = 4AV (Lemma 3 to Prop.
2), and QD = ±A0-AV=AV. But
H divides QD in the same ratio as G
divides A (Prop. 7) ; therefore
VK = QH = m.QD = m.AV.
Taking moments about A of the segment, the triangle ABB'
and the sum of the small segments, we have (dividing out by
A V and A ABB')
§(1 + m) + §. 4 = -1 .4m,
or 15 m = 9,
and m = ■§ .
That is, AG = %A0, or AG : GO = 3 : 2.
The final proposition (10) finds the centre of gravity of the
portion of a parabola cut off between two parallel chords PP',
BB'. If PP' is the shorter of the chords and the diameter
bisecting PP', BB' meets them in N, respectively, Archi-
medes proves that, if NO be divided into five equal parts of
which LM is the middle one (L being nearer to N than M is),
ON PLANE EQUILIBRIUMS, II 81
the centre of gravity G of the portion of the parabola between
PP' and BB' divides LM in such a way that
LG:GM=BO*.(2PN+BO):PN 2 .(2BO + PN).
The geometrical proof is somewhat difficult, and uses a very
remarkable Lemma which forms Prop. 9. If a, b, c, d, x, y are
straight lines satisfying the conditions
a b c , \
T = - = -j (a > o > c > a),
b c a v '
d x
a — d i( a ~ c )
2a + 4:b + 6c + 3d y
and
5a+ 106 + 10c + 5(i a — c
then must x + y = -fa.
The proof is entirely geometrical, but amounts of course to
the elimination of three quantities b, c, d from the above four
equations.
The Sand-reckoner (Psammites or Arenarhis).
I have already described in a previous chapter the remark-
able system, explained in this treatise and in a lost work,
'Apxai, Principles, addressed to Zeuxippus, for expressing very
large numbers which were beyond the range of the ordinary
Greek arithmetical notation. Archimedes showed that his
system would enable any number to be expressed up to that
which in our notation would require 80,000 million million
ciphers and then proceeded to prove that this system more
than sufficed to express the number of grains of sand which
it would take to fill the universe, on a reasonable view (as it
seemed to him) of the size to be attributed to the universe.
Interesting as the book is for the course of the argument by
which Archimedes establishes this, it is, in addition, a docu-
ment of the first importance historically. It is here that we
learn that Aristarchus put forward the Copernican theory of
the universe, with the sun in the centre and the planets
including the earth revolving round it, and that Aristarchus
further discovered the angular diameter of the sun to be yjo^h
of the circle of the zodiac or half a degree. Since Archimedes,
in order to calculate a safe figure (not too small) for the size
1523.2 Q
82 ARCHIMEDES
of the universe, has to make certain assumptions as to the
sizes and distances of the sun and moon and their relation
to the size of the universe, he takes the opportunity of
quoting earlier views. Some have tried, he says, to prove
that the perimeter of the earth is about 300,000 stades; in
order to be quite safe he will take it to be about ten times
this, or 3,000,000 stades, and not greater. The diameter of
the earth, like most earlier astronomers, he takes to be
greater than that of the moon but less than that of the sun.
Eudoxus, he says, declared the diameter of the sun to be nine
times that of the moon, Phidias, his own father, twelve times,
while Aristarchus tried to prove that it is greater than 18 but
less than 20 times the diameter of the moon; he will again be
on the safe side and take it to be 30 times, but not more. The
position is rather more difficult as regards the ratio of the
distance of the sun to the size of the universe. Here he seizes
upon a dictum of Aristarchus that the sphere of the fixed
stars is so great that the circle in which he supposes the earth
to revolve (round the sun) ' bears such a proportion to the
distance of the fixed stars as the centre of the sphere bears to
its surface '. If this is taken in a strictly mathematical sense,
it means that the sphere of the fixed stars is infinite in size,
which would not suit Archimedes's purpose ; to get another
meaning out of it he presses the point that Aristarchus's
words cannot be taken quite literally because the centre, being
without magnitude, cannot be in any ratio to any other mag-
nitude ; hence he suggests that a reasonable interpretation of
the statement would be to suppose that, if we conceive a
sphere with radius equal to the distance between the centre
of the sun and the centre of the earth, then
(diam. of earth) : (diam. of said sphere)
= (diam. of said sphere) : (diam. of sphere of fixed stars).
This is, of course, an arbitrary interpretation ; Aristarchus
presumably meant no such thing, but merely that the size of
the earth is negligible in comparison with that of the sphere
of the fixed stars. However, the solution of Archimedes's
problem demands some assumption of the kind, and, in making
this assumption, he was no doubt aware that he was taking
a liberty with Aristarchus for the sake of giving his hypo-
thesis an air of authority.
THE SAND-RECKONER
83
Archimedes has, lastly, to compare the diameter of the sun
with the circumference of the circle described by its centre.
Aristarchus had made the apparent diameter of the sun ylo^h
of the said circumference ; Archimedes will prove that the
said circumference cannot contain as many as 1,000 sun's
diameters, or that the diameter of the sun is greater than the
side of a regular chiliagon inscribed in the circle. First he
made an experiment of his own to determine the apparent
diameter of the sun. With a small cylinder or disc in a plane
at right angles to a long straight stick and moveable along it,
he observed the sun at the moment when it cleared the
horizon in rising, moving the disc till it just covered and just
failed to cover the sun as he looked along the straight stick.
He thus found the angular diameter to lie between T ^ ^R and
^oR, where R is a right angle. But as, under his assump-
tions, the size of the earth is not negligible in comparison with
the sun's circle, he had to allow for parallax and find limits
for the angle subtended by the sun at the centre of the earth.
This he does by a geometrical argument very much in the
manner of Aristarchus.
Let the circles with centres 0, G represent sections of the sun
and earth respectively, E the position of the observer observing
g2
84 ARCHIMEDES
the sun when it has just cleared the horizon. Draw from E
two tangents EP, EQ to the circle with centre 0, and from
C let CF, CG be drawn touching the same circle. With centre
G and radius GO describe a circle : this will represent the path
of the centre of the sun round the earth. Let this circle meet
the tangents from G in A, B, and join A B meeting GO in M.
Archimedes's observation has shown that
?b f B> I PEQ >^ B;
and he proceeds to prove that AB is less than the side of a
regular polygon of 656 sides inscribed in the circle AOB,
but greater than the side of an inscribed regular polygon of
1,000 sides, in other words, that
T ^R >IFCG > aio^-
The first relation is obvious, for, since GO > E0,
Z PEQ > Z FGG.
Next, the perimeter of any polygon inscribed in the circle
AOB is less than - 4 T 4 - GO (i.e. -~- times the diameter) ;
Therefore AB < ^ • -\ 4 - CO or T ^g CO,
and, a fortiori, AB < t Jq CO.
Now, the triangles GAM, COF being equal in all respects,
AM= OF, so that AB = 20F= (diameter of sun) > CH+ OK,
since the diameter of the sun is greater than that of the earth ;
therefore CH + OK < ^CO, and HK > ^CO.
And CO > CF, while HK < EQ, so that EQ > T %%CF.
We can now compare the angles OCF, OEQ ;
tan OCF-
Z OCF
for TOEQ
>
tan 0EQ_
EQ
> CF
> T 9 o 9 o, a fortiori.
Doubling the angles, we have
IFGG >^%.IPEQ
> 2omo^ since I PEQ > ^i?,
■^ 203 ■ Z1 '*
THE SAND-RECKONER 85
Hence AB is greater than the side of a regular polygon of
812 sides, and a fortiori greater than the side of a regular
polygon of 1,000 sides, inscribed in the circle AOB.
The perimeter of the chiliagon, as of any regular polygon
with more sides than six, inscribed in the circle A OB is greater
than 3 times the diameter of the sun's orbit, but is less than
1,000 times the diameter of the sun, and a fortiori less than
30,000 times the diameter of the earth;
therefore (diameter of sun's orbit) < 10,000 (diam. of earth)
< 10,000,000,000 stades.
But (diam. of earth) : (diam. of sun's orbit)
= (diam. of sun's orbit) : (diam. of universe) ;
therefore the universe, or the sphere of the fixed stars, is less
than 10,000 3 times the sphere in which the sun's orbit is a
great circle.
Archimedes takes a quantity of sand not greater than
a poppy-seed and assumes that it contains not more than 10,000
grains ; the diameter of a poppy-seed he takes to be not less
than 4 X oth of a finger-breadth ; thus a sphere of diameter
1 finger-breadth is not greater than 64,000 poppy-seeds and
therefore contains not more than 640,000,000 grains of sand
('6 units of second order + 40,000,000 units of first order')
and a fortiori not more than 1,000,000,000 ('10 units of
second order of numbers '). Gradually increasing the diameter
of the sphere by multiplying it each time by 100 (making the
sphere 1,000,000 times larger each time) and substituting for
10,000 finger-breadths a stadium (< 10,000 finger-breadths),
he finds the number of grains of sand in a sphere of diameter
10,000,000,000 stadia to be less than '1,000 units of seventh
order of numbers ' or 10 51 , and the number in a sphere 10,000 3
times this size to be less than ' 10,000,000 units of the eighth
order of numbers ' or 1 63 .
The Quadrature of the Parabola.
In the preface, addressed to Dositheus after the death of
Conon, Archimedes claims originality for the solution of the
problem of finding the area of a segment of a parabola cut off
by any chord, which he says he first discovered by means of
mechanics and then confirmed by means of geometry, using
the lemma that, if there are two unequal areas (or magnitudes
86 ARCHIMEDES
generally), then however small the excess of the greater over
the lesser, it can by being continually added to itself be made
to exceed the greater ; in other words, he confirmed the solution
by the method of exhaustion. One solution by means of
mechanics is, as we have seen, given in The Method ; the
present treatise contains a solution by means of mechanics
confirmed by the method of exhaustion (Props. 1-17), and
then gives an entirely independent solution by means of pure
geometry, also confirmed by exhaustion (Props. 18-24).
I. The mechanical solution depends upon two properties of
the parabola proved in Props. 4, 5. If Qq be the base, and P
the vertex, of a parabolic segment, P is the point of contact
of the tangent parallel to Qq, the diameter PV through P
bisects Qq in V, and, if VP produced meets the tangent at Q
in T, then TP = PV. These properties, along with the funda-
mental property that QV 2, varies as PV, Archimedes uses to
prove that, if EO be any parallel to TV meeting QT, QP
(produced, if necessary), the curve, and Qq in E, F, R,
respectively, then
QV:VO = OF:FR,
and QO : Oq =ER: RO. (Props. 4, 5.)
Now suppose a parabolic segment QR X q so placed in relation
to a horizontal straight line QA through Q that the diameter
bisecting Qq is at right angles to QA, i.e. vertical, and let the
tangent at Q meet the diameter qO through q in E. Produce
QO to A, making OA equal to OQ.
Divide Qq into any number of equal parts at X , 2 . . . n ,
and through these points draw parallels to OE, i.e. vertical
lines meeting OQ in H l , H 2 , ..., EQ in E 1} E 2 , ..., and the
THE QUADRATURE OF THE PARABOLA 87
curve in E 1 , R 2 , ... . Join QR X , and produce it to meet OE in
F, QB 2 meeting l E l in F lt and so on.
o Ht h 2 h 3 Hy, a
Now Archimedes has proved in a series of propositions
(6-13) that, if a trapezium such as 1 E 1 E 2 2 is suspended
from H X H 2 , and an area P suspended at J. balances 1 E 1 E 2 2
so suspended, it will take a greater area than P suspended at
A to balance the same trapezium suspended from H 2 and
a less area than P to balance the same trapezium suspended
from H 1 . A similar proposition holds with regard to a triangle
such as E n H n Q suspended where it is and suspended at Q and
H n respectively.
Suppose (Props. 14, 15) the triangle QqE suspended where
it is from OQ, and suppose that the trapezium E0 lt suspended
where it is, is balanced by an area F^ suspended at A, the
trapezium E l 2 , suspended where it is, is balanced by P 2
suspended at A, and so on, and finally the triangle E n O n Q,
suspended where it is, is balanced by P n+1 suspended at A ;
then P Y + P 2 + ... +P n+1 at A balances the whole triangle, so that
P 1 + P 2 +... + P n+l = iA_E q Q,
since the whole triangle may be regarded as suspended from
the point on OQ vertically above its centre of gravity.
Now AO:OH l = QO:OH l
= Qq:q0 1
= E^O^O^, by Prop. 5,
= (trapezium EO^) : (trapezium P0 2 ),
88 ARCHIMEDES
that is, it takes the trapezium F0 Y suspended at A to balance
the trapezium E0 1 suspended at H v And P 2 balances E0 X
where it is.
Therefore (FO,) > P 2 .
Similarly (^1^2) > ^2' anc ^ so on -
Again AO :0H 1 = E^: X R X
= (trapezium E 1 2 ) : (trapezium P 2 2 ),
that is, {R x 2 ) at A will balance (E 1 2 ) suspended at H 1 ,
while P 2 at A balances (Efi^) suspended where it is,
whence P 2 > R Y 2 .
Therefore (F x 0^ > P 2 > (PjOJ,
(P 2 3 ) > P 6 > P 2 3 , and so on;
and finally, &H n °nQ > P n+i > & R n °nQ'
By addition,
(R 1 2 )+(R 2 O i ) + ...+(AR n O n Q)<P 2 + P + ...+P n+l ;
therefore, a fortiori,
(R 1 0^ + (R. z 3 ) + ... + AR n n Q<P 1 +P ! +...+P n+1
That is to say, we have an inscribed figure consisting of
trapezia and a triangle which is less, and a circumscribed
figure composed in the same way which is greater, than
P 1 + P 2 + ...+P n+1 , i.e. iAEqQ.
It is therefore inferred, and proved by the method of ex-
haustion, that the segment itself is equal to ^AEqQ (Prop. 16).
In order to enable the method to be applied, it has only
to be proved that, by increasing the number of parts in Qq
sufficiently, the difference between the circumscribed and
inscribed figures can be made as small as we please. This
can be seen thus. We have first to show that all the parts, as
qF, into which qE is divided are equal.
We have E 1 1 : 0^ = QO : 0H X = (ra+ 1) : 1,
or 0,R, = .E,0,, whence also 9 S = -— . 9 E 9 .
11 n+\ Y l 2 w+1 22
THE QUADRATURE OF THE PARABOLA 89
And E 2 2 : 2 R 2 = QO : 0H 2 = (n + 1) : 2,
or 2 R 2 =—-.0 2 E 2 .
It follows that 2 S = SR 2 , and so on.
Consequently 1 R 1 , 2 R 2 , 3 R 3 ... are divided into 1,2, 3 ...
equal parts respectively by the lines from Q meeting qE.
It follows that the difference between the circumscribed and
inscribed figures is equal to the triangle FqQ, which can be
made as small as we please by increasing the number of
divisions in Qq, i.e. in qE.
Since the area of the segment is equal to J A Eq Q, and it is
easily proved (Prop. 17) that AEqQ = 4 (triangle with same
base and equal height with segment), it follows that the area
of the segment = § times the latter triangle.
It is easy to see that this solution is essentially the same as
that given in The Method (see pp. 29-30, above), only in a more
orthodox form (geometrically speaking). For there Archi-
medes took the sum of all the straight lines, as 1 R 1 , 2 R 2 ... ,
as making up the segment notwithstanding that there are an
infinite number of them and straight lines have no breadth.
Here he takes inscribed and circumscribed trapezia propor-
tional to the straight lines and having finite breadth, and then
compresses the figures together into the segment itself by
increasing indefinitely the number of trapezia in each figure,
i.e. diminishing their breadth indefinitely.
The procedure is equivalent to an integration, thus :
If X denote the area of the triangle FqQ, we have, if n be
the number of parts in Qq,
(circumscribed figure)
= sum of As QqF, QR X F^ QR 2 F 2 , ...
= sum of AsQqF, QO x R lt Q0 2 S, ...
(n- 1) 2 (^-2) 2
1
= -™. X(Z 2 + 2 2 Z 2 + 3 a Z 2 + ... + r^Z 2 ).
Similarly, we find that
(inscribed figure) = -^-^ X {X 2 + 2 2 X 2 + ... + (n- 1) 2 X 2 }.
lb A.
90
ARCHIMEDES
Taking the limit, wc have, if A denote the area of the
triangle EqQ, so that A = nX,
area of segment
1
'"J 2
= IA.
X 2 dX
II. The purely geometrical method simply exhausts the
parabolic segment by inscribing successive figures ' in the
recognized manner' (see p. 79, above). For this purpose
it is necessary to find, in terms of the triangle with the same
base and height, the area added to the
inscribed figure by doubling the number of
sides other than the base of the segment.
Let QPq be the triangle inscribed ' in the
recognized manner ', P being the point of
contact of the tangent parallel to Qq, and
PV the diameter bisecting Qq. If QV, Vq
be bisected in M, m, and RM, rm be drawn
parallel to PV meeting the curve in R, r,
the latter points are vertices of the next
figure inscribed ' in the recognized manner ',
for RY, ry are diameters bisecting PQ, Pq
respectively.
4RW 2 , so that PV = 4PW, or RM = 3PJT.
YM=±PV
2PW, so that YM =2RY.
APRQ = ±APQM=±APQV.
Now QV 2
But
Therefore
Similarly
APrq = ±APVq; whence (APRQ + APrq)= ±PQq. (Prop. 21.)
In like manner it can be proved that the next addition
to the inscribed figure adds J of the sum of AsPRQ, Prq,
and so on.
Therefore the area of the inscribed figure
= [l+i+a) 2 + ...}.APQg. (Prop. 22.)
Further, each addition to the inscribed figure is greater
than half the segments of the parabola left over before the
addition is made. For, if we draw the tangent at P and
complete the parallelogram EQqe with side EQ parallel to PV,
THE QUADRATURE OF THE PARABOLA 91
the triangle PQq is half of the parallelogram and therefore
more than half the segment. And so on (Prop. 20).
We now have to sum n terms of the above geometrical
series. Archimedes enunciates the problem in the form, Given
a series of areas A, B, C, D . . . Z, of which A is the greatest, and
each is equal to four times the next in order, then (Prop. 23)
A+B + C+... + Z+iZ = §A.
The algebraical equivalent of this is of course
i+H(i) 2 +...+(ir l =f-iar i = :! f 1 ^-
1 4
To find the area of the segment, Archimedes, instead of
taking the limit, as we should, uses the method of reductio ad
absurdum.
Suppose K — f . A PQq.
(1) If possible, let the area of the segment be greater than K. ,
We then inscribe a figure ' in the recognized manner ' such
that the segment exceeds it by an area less than the excess of
the segment over K. Therefore the inscribed figure must be
greater than K, which is impossible since
A + B + C+...+Z< §4,
where A = APQq (Prop. 23).
(2) If possible, let the area of the segment be less than K.
If then APQq = A, B = \A, G = \B, and so on, until we
arrive at an area X less than the excess of K over the area of
the segment, we have
A + B + C+ ... +X + iX = %A = K.
Thus K exceeds A + B + C+ ... + X by an area less than X,
and exceeds the segment by an area greater than X.
It follows that A +B + C+ ... +X> (the segment) ; which
is impossible (Prop. 22).
Therefore the area of the segment, being neither greater nor
less than K, is equal to K or f APQq.
On Floating Bodies, I, II.
In Book I of this treatise Archimedes lays down the funda-
mental principles of the science of hydrostatics. These are
92 ARCHIMEDES
deduced from Postulates which are only two in number. The
first which begins Book I is this :
1 let it be assumed that a fluid is of such a nature that, of the
parts of it which lie evenly and are continuous, that which is
pressed the less is driven along by that which is pressed the
more ; and each of its parts is pressed by the fluid which is
perpendicularly above it except when the fluid is shut up in
anything and pressed by something else ' ;
the second, placed after Prop. 7, says
' let it be assumed that, of bodies which are borne upwards in
a fluid, each is borne upwards along the perpendicular drawn
through its centre of gravity \
Prop. 1 is a preliminary proposition about a sphere, and
then Archimedes plunges in medias res with the theorem
(Prop. 2) that ' the surface of any fluid at rest is a sphere the
centre of which is the same as that of the earth ', and in the
whole of Book I the surface of the fluid is always shown in
the diagrams as spherical. The method of proof is similar to
what we should expect in a modern elementary textbook, the
main propositions established being the following. A solid
which, size for size, is of equal weight with a fluid will, if let
down into the fluid, sink till it is just covered but not lower
(Prop. 3) ; a solid lighter than a fluid will, if let down into it,
be only partly immersed, in fact just so far that the weight
of the solid is equal to the weight of the fluid displaced
(Props. 4, 5), and, if it is forcibly immersed, it will be driven
upwards by a force equal to the difference between its weight
and the weight of the fluid displaced (Prop. 6).
The important proposition follows (Prop. 7) that a solid
heavier than a fluid will, if placed in it, sink to the bottom of
the fluid, and the solid will, when weighed in the fluid, be
lighter than its true weight by the weight of the fluid
displaced.
The problem of the Crown.
This proposition gives a method of solving the famous
problem the discovery of which in his bath sent Archimedes
home naked crying tvprjKa, evprjKa, namely the problem of
ON FLOATING BODIES, I 93
determining the proportions of gold and silver in a certain
crown.
Let W be the weight of the crown, w 1 and tv 2 the weights of
the gold and silver in it respectively, so that W = w x + w 2 .
(1) Take a weight IT of pure gold and weigh it in the fluid.
The apparent loss of weight is then equal to the weight of the
fluid displaced ; this is ascertained by weighing. Let it be F v
It follows that the weight of the fluid displaced by a weight
w i °^ gold is -=^ . F r
(2) Take a weight W of silver, and perform the same
operation. Let the weight of the fluid displaced be F 2 .
Then the weight of the fluid displaced by a weight w 2 of
silver is ^S> F .
(3) Lastly weigh the crown itself in the fluid, and let F be
loss of weight or the weight of the fluid displaced.
We have then ^ . F x + ^ . F„ = F,
that is, w 1 F x + w 2 F 2 = (w x + w 2 ) F,
, w, F 2 -F
whence — * = -=^ — r^--
w 2 F-F x
According to the author of the poem de 'ponderibus et men-
surls (written probably about a.d. 500) Archimedes actually
used a method of this kind. We first take, says our authority,
two equal weights of gold and silver respectively and weigh
them against each other when both are immersed in water ;
this gives the relation between their weights in water, and
therefore between their losses of weight in water. Next we
take the mixture of gold and silver and an equal weight of
silver, and weigh them against each other in water in the
same way.
Nevertheless I do not think it probable that this was the
way in which the solution of the problem was discovered. As
we are told that Archimedes discovered it in his bath, and
that he noticed that, if the bath was full when he entered it,
so much water overflowed as was displaced by his body, he is
more likely to have discovered the solution by the alternative
94 ARCHIMEDES
method attributed to him by Vitruvius, 1 namely by measuring
successively the volumes of fluid displaced by three equal
weights, (1) the crown, (2) an equal weight of gold, (3) an
equal weight of silver respectively. Suppose, as before, that
the weight of the crown is W and that it contains weights
tu 1 and iv 2 of gold and silver respectively. Then
(1) the crown displaces a certain volume of the fluid, V, say ;
(2) the weight W of gold displaces a volume V v say, of the
fluid ;
therefore a weight w x of gold displaces a volume yiy- V x of
the fluid ;
(3) the weight W of silver displaces V 2 , say, of the fluid;
w
therefore a weight w 2 of silver displaces —• V 2 .
It follows that V = ^ • V 1 + ^ • V 2 ,
whence we derive (since W = w 1 + w 2 )
y\ v 2 -v
w 2 ~ V-V]'
the latter ratio being obviously equal to that obtained by the
other method.
The last propositions (8 and 9) of Book I deal with the case
of any segment of a sphere lighter than a fluid and immersed
in it in such a way that either (1) the curved surface is down-
wards and the base is entirely outside the fluid, or (2) the
curved surface is upwards and the base is entirely submerged,
and it is proved that in either case the segment is in stable
equilibrium when the axis is vertical. This is expressed here
and in the corresponding propositions of Book II by saying
that, ' if the figure be forced into such a position that the base
of the segment touches the fluid (at one point), the figure will
not remain inclined but will return to the upright position '.
Book II, which investigates fully the conditions of stability
of a right segment of a paraboloid of revolution floating in
a fluid for different values of the specific gravity and different
ratios between the axis or height of the segment and the
1 De architectural, ix. 3.
ON FLOATING BODIES, I, II 95
principal parameter of the generating parabola, is a veritable
tour de force which must be read in full to be appreciated.
Prop. 1 is preliminary, to the effect that, if a solid lighter than
a fluid be at rest in it, the weight of the solid will be to that
of the same volume of the fluid as the immersed portion of
the solid is to the whole. The results of the propositions
about the segment of a paraboloid may be thus summarized.
Let h be the axis or height of the segment, p the principal
parameter of the generating parabola, s the ratio of the
specific gravity of the solid to that of the fluid (s always < 1 ).
The segment is supposed to be always placed so that its base
is either entirely above, or entirely below, the surface of the
fluid, and what Archimedes proves in each case is that, if
the segment is so placed with its axis inclined to the vertical
at any angle, it will not rest there but will return to the
position of stability.
I. If h is not greater than §p, the position of stability is with
the axis vertical, whether the curved surface is downwards or
upwards (Props. 2, 3).
II. If h is greater than f p, then, in order that the position of
stability may be with the axis vertical, s must be not t less
than (h — ^pY/h? if the curved surface is downwards, and not
greater than {h 2 — (h — %p) 2 }/h 2 if the curved surface is
upwards (Props. 4, 5).
III. If h>%p, but h/^p < 15/4, the segment, if placed with
one point of the base touching the surface, will never remain
there whether the curved surface be downwards or upwards
(Props. 6, 7). (The segment will move in the direction of
bringing the axis nearer to the vertical position.)
IV. If h>ip, but k/±p<\5/ 4, and if s is less than
(h — ^pf/Ti 2 in the case where the curved surface is down-
wards, but greater than {A 2 --(& — J^) 2 }/^ 2 i n the case where
the curved surface is upwards, then the position of stability is
one in which the axis is not vertical but inclined to the surface
of the fluid at a certain angle (Props. 8, 9). (The angle is drawn
in an auxiliary figure. The construction for it in Prop. 8 is
equivalent to the solution of the following equation in 0,
\p cot 2 # s= |(/i — &) — |p,
96
ARCHIMEDES
where k is the axis of the segment of the paraboloid cut off by
the surface of the fluid.)
V. Prop. 10 investigates the positions of stability in the cases
where h/±p> 15/4, the base is entirely above the surface, and
« has values lying between five pairs of ratios respectively.
Only in the case where s is not less than (h — ^pf/h 2 is the
position of stability that in which the axis is vertical.
BAB 1 is a section of the paraboloid through the axis AM.
G is a point on AM such that AG = 2 CM, K is a point on GA
such that AM-.CK =15:4. CO is measured along GA such
that GO = %p, and R is a point on AM such that MR = §C0.
A 2 is the point in which the perpendicular to AM from K
meets AB, and A 3 is the middle point of AB. BA 2 B 2 , BA Z M
are parabolic segments on A 2 M 2 , A 3 M 3 (parallel to AM) as axes
and similar to the original segment. (The parabola BA.,B 2
is proved to pass through G by using the above relation
AM: GK =15:4 and applying Prop. 4 of the Quadrature of
the Parabola.) The perpendicular to AM from meets the
parabola BA 2 B 2 in two points P 2 , Q 2 , and straight lines
through these points parallel to AM meet the other para-
bolas in P-p Q 1 and P 3 , Q 3 respectively. P X T and Q 1 U are
tangents to the original parabola meeting the axis MA pro-
duced in T, U. Then
(i) if s is not less than AR 2 :AM 2 or (h — ^p) 2 :h 2 , there is
stable equilibrium when AM is vertical ;
THE CATTLE-PROBLEM 97
(ii) if s<AR 2 :AM 2 but >Q 1 Q 2 :AM\ the solid will not rest
with its base touching the surface of the fluid in one point
only, but in a position with the base entirely out of the fluid
and the axis making with the surface an angle greater
than U ;
(iiia) if s = Q X Q 2 \ AM 2 , there is stable equilibrium with one
point of the base touching the surface and AM inclined to it
at an angle equal to U;
(iiib) if s = P l P 5 2 : AM 2 , there is stable equilibrium with one
point of the base touching the surface and with AM inclined
to it at an angle equal to T ;
(iv) iis>P 1 P 2 :AM 2 but <Q X Q 2 :AM 2 , there will be stable
equilibrium in a position in which the base is more submerged ;
(v) if s<P x P 2 : AM 2 , there will be stable equilibrium with
the base entirely out of the fluid and with the axis AM
inclined to the surface at an angle less than T.
It remains to mention the traditions regarding other in-
vestigations by Archimedes which have reached us in Greek
or through the Arabic.
(a) The Cattle-Problem.
This is a difficult problem in indeterminate analysis. It is
required to find the number of bulls and cows of each of four
colours, or to find 8 unknown quantities. The first part of
the problem connects the unknowns by seven simple equations ;
and the second part adds two more conditions to which the
unknowns must be subject. Ii W, iv be the numbers of white
bulls and cows respectively and (X, x), (F, y), (Z, z) represent
the numbers of the other three colours, we have first the
following equations :
(I) Tf=(4 + |)Z + F, (a) ,
X = ($ + i)Z+Y, (/?)
Z^d + ftW+Y, (y)
(II) w = ft + $(!+»), (S)
x=(k + k){Z+z), («)
« = (*+*) (7+y). W
y = (i+k)(W+w). (,)
1523.2 H
98 ARCHIMEDES
Secondly, it is required that
W+X = a square, (6)
Y+Z — a triangular number. (i)
There is an ambiguity in the text which makes it just possible
that W+ X need only be the product of two whole numbers
instead of a square as in (0). Jul. Fr. Wurm solved the problem
in the simpler form to which this change reduces it. The
complete problem is discussed and partly solved by Amthor. 1
The general solution of the first seven equations is
W= 2.3.7.53.4657?! = 1036648271,
X = 2. 3 2 . 89. 465771 = 7460514ft,
Y= 3 4 . 11 .465771 = 414938771,
Z- 2 2 . 5.79.465771 == 735806071,
w- 2 3 . 3. 5. 7.23. 37371= 720636071,
X = 2.3M7. 1599171 = 48932467*,
y = 3 2 . 13.4648971 = 543921371,
z = 2 2 . 3. 5. 7.11.76171 = 351582071.
It is not difficult to find such a value of n that W+ X = a
square number; it is n = 3 . 11 . 29 . 4657£ 2 = 4456749£ 2 ,
where £ is any integer. We then have to make Y + Z
a triangular number, i.e. a number of the form i<7(#+l).
This reduces itself to the solution of the ' Pellian ' equation
£ 2 -4729494u 2 = 1,
which leads to prodigious figures ; one of the eight unknown
quantities alone would have more than 206,500 digits!
(/?) On semi-regular polyhedra.
In addition, Archimedes investigated polyhedra of a certain
type. This we learn from Pappus. 2 The polyhedra in question
are semi-regular, being contained by equilateral and equi-
1 Zeitschrift fur Math. u. Fhysik (Hist.-litt. Abt.) xxv. (1880), pp.
156 sqq.
2 Pappus, v, pp. 352-8.
ON SEMI-REGULAR POLYHEDRA 99
angular, but not similar, polygons ; those discovered by
Archimedes were 13 in number. If we for convenience
designate a polyhedron contained by m regular polygons
of oc sides, n regular polygons of /? sides, &c, by (m a , %...),
the thirteen Archimedean polyhedra, which we will denote by
jF}, P 2 ...P IZ , are as follows:
Figure with 8 faces: P x = (4,, 4 G ).
Figures with 14 faces: P 2 = (8 3 , 6 4 ), P 3 = (6 4 , 8 6 ),
P 4 = (8 3J 6 8 ).
Figures with 26 faces : P 5 = (8 3 , 18 4 ), P 6 = (12 4 , 8 6 , 6 8 ).
Figures with 32 faces: P 7 = (20 3 , 12 5 ), P 8 = (12 5 , 20 6 ),
P 9 = (20 lf 12 10 ).
Figure with 38 faces: P 10 = (32 3 , 6 4 ).
Figures with 62 faces: P n = (20 3 , 30 4 , 12 5 ),
P ]2 EE(30 4 ,20 G ,12 10 ).
Figure with 92 faces: P 13 = (80 3 , 12 5 ).
Kepler 1 showed how these figures can be obtained. A
method of obtaining some of them is indicated in a fragment
of a scholium to the Vatican MS. of Pappus. If a solid
angle of one of the regular solids be cut off symmetrically by
a plane, i.e. in such a way that the plane cuts off the same
length from each of the edges meeting at the angle, the
section is a regular polygon which is a triangle, square or
pentagon according as the solid angle is formed of three, four,
or five plane angles. If certain equal portions be so cut off
from all the solid angles respectively, they will leave regular
polygons inscribed in the faces of the solid ; this happens
(A) when the cutting planes bisect the sides of the faces and
so leave in each face a polygon of the same kind, and (B) when
the cutting planes cut off a smaller portion from each angle in
such a way that a regular polygon is left in each face which
has double the number of sides (as when we make, say, an
octagon out of a square by cutting off the necessary portions,
1 Kepler, Harmonice mundi in Opera (1864), v, pp. 123-6.
H 2
100
ARCHIMEDES
symmetrically, from the corners). We have seen that, accord-
ing to Heron, two of the semi-regular solids had already been
discovered by Plato, and this would doubtless be his method.
The methods (A) and (B) applied to the five regular solids
give the following out of the 13 semi-regular solids. We
obtain (1) from the tetrahedron, P 1 by cutting off angles
so as to leave hexagons in the faces ; (2) from the cube, P 2 by
leaving squares, and P 4 by leaving octagons, in the faces ;
(3) from the octahedron, P 2 by leaving triangles, and P 3 by
leaving hexagons, in the faces ; (4) from the icosahedron,
Bj by leaving triangles, and P g by leaving hexagons, in the
faces; (5) from the dodecahedron, P 7 by leaving pentagons,
and P 9 by leaving decagons in the faces.
Of the remaining six, four are obtained by cutting off all
the edges symmetrically and equally by planes parallel to the
edges, and then cutting off angles. Take first the cube.
(1) Cut off from each four parallel edges portions which leave
an octagon as the section of the figure perpendicular to the
edges ; then cut off equilateral triangles from the corners
(see Fig. 1) ; this gives P 5 containing 8 equilateral triangles
and 18 squares. (P 5 is also obtained by bisecting all the
edges of P 2 and cutting off corners.) (2) Cut off from the
edges of the cube a smaller portion so as to leave in each
face a square such that the octagon described in it has its
side equal to the breadth of the section in which each edge is
cut; then cut off hexagons from each angle (see Fig. 2); this
""r "jr "" \*Y~
• ■
! , l—
Fig. 1.
Fig. 2.
gives 6 octagons in the faces, 12 squares under the edges and
8 hexagons at the corners; that is, we have P 6 . An exactly
ON SEMI-REGULAR' POLYHEDRA
101
similar procedure with the icosahedron and dodecahedron
produces P n and P l2 (see Figs. 3, 4 for the case of the icosa-
hedron).
Fig. 3.
Fig. 4.
The two remaining solids P 10 , P 13 cannot be so simply pro-
duced. They are represented in Figs. 5, 6, which I have
Fig. 5.
Fig. 6.
taken from Kepler. P l0 is the snub cube in which each
solid angle is formed by the angles of four equilateral triangles
and one square; P 13 is the snub dodecahedron, each solid
angle of which is formed by the angles of four equilateral
triangles and one regular pentagon.
We are indebted to Arabian tradition for
(y) The Liber Assumptorum.
Of the theorems contained in this collection many are
so elegant as to afford a presumption that they may really
be due to Archimedes. In three of them the figure appears
which was called dpftrjXos, a shoemaker's knife, consisting of
three semicircles with a common diameter as shown in the
annexed figure. If N be the point at which the diameters
102
ARCHIMEDES
of the two smaller semicircles adjoin, and NP be drawn at
right angles to AB meeting the external semicircle in P, the
area of the apfi-qXos (included between the three semicircular
arcs) is equal to the circle on PN as diameter (Prop. 4). In
Prop. 5 it is shown that, if a circle be described in the space
between the arcs AP, AN and the straight line PN touching
all three, and if a circle be similarly described in the space
between the arcs PB, NB and the straight line PN touching
all three, the two circles are equal. If one circle be described
in the dpftrjXos touching all three semicircles, Prop. 6 shows
that, if the ratio of AN to NB be given, we can find the
relation between the diameter of the circle inscribed to the
dpftrjXos and the straight line AB ; the proof is for the parti-
cular case AN = §BN, and shows that the diameter of the
inscribed circle = -f^AB.
Prop. 8 is of interest in connexion with the problem of
trisecting any angle. If AB be any chord of a circle with
centre 0, and BC on AB produced be made equal to the radius,
draw CO meeting the circle in D, E ; then will the arc BD be
one-third of the arc AE (or BF, if EF be the chord through E
parallel to AB). The problem is by this theorem reduced to
a v ever is (cf. vol. i, p. 241).
THE LIBER ASSUMPTORUM
103
Lastly, we may mention the elegant theorem about the
area of the Salinon (presumably ' salt-cellar ') in Prop. 14.
ACB is a semicircle on AB as diameter, AD, EB are equal
lengths measured from A and B on AB. Semicircles are
drawn with AD, EB as diameters on the side towards G, and
a semicircle with DE as diameter is drawn on the other side of
AB. CF is the perpendicular to A B through 0, the centre
of the semicircles ACB, DFE. Then is the area bounded by
all the semicircles (the Salinon) equal to the circle on CF
as diameter.
The Arabians, through whom the Book of Lemmas has
reached us, attributed to Archimedes other works (1) on the
Circle, (2) on the Heptagon in a Circle, (3) on Circles touch-
ing one another, (4) on Parallel Lines, (5) on Triangles, (6) on
the properties of right-angled triangles, (7) a book of Data,
(8) De clepsydris : statements which we are not in a position
to check. But the author of a book on the finding of chords
in a circle, 1 Abu'l Raihan Muh. al-Biruni, quotes some alterna-
tive proofs as coming from the first of these works.
(8) Formula for area of triangle.
More important, however, is the mention in this same work
of Archimedes as the discoverer of two propositions hitherto
attributed to Heron, the first being the problem of finding
the perpendiculars of a triangle when the sides are given, and
the second the famous formula for the area of a triangle in
terms of the sides,
V{s(s — a)(s — b) (s — c)}.
1 See Bibliotheca mathematica, xi 3 , pp. 11-78.
104 ERATOSTHENES
Long as the present chapter is, it is nevertheless the most
appropriate place for Eratosthenes of Cyrene. It was to him
that Archimedes dedicated The Method, and the Cattle-Problem
purports, by its heading, to have been sent through him to
the mathematicians of Alexandria. It is evident from the
preface to The Method that Archimedes thought highly of his
mathematical ability. He was, indeed, recognized by his con-
temporaries as a man of great distinction in all branches of
knowledge, though in each subject he just fell short of the
highest place. On the latter ground he was called Beta, and
another nickname applied to him, Pentathlos, has the same
implication, representing as it does an all-round athlete who
was not the first runner or wrestler but took the second prize
in these contests as well as in others. He was very little
younger than Archimedes ; the date of his birth was probably
284 b.c. or thereabouts. He was a pupil of the philosopher
Ariston of Chios, the grammarian Lysanias of Cyrene, and
the poet Callimachus ; he is said also to have been a pupil of
Zeno the Stoic, and he may have come under the influence of
Arcesilaus at Athens, where he spent a considerable time.
Invited, when about 40 years of age, by Ptolemy Euergetes
to be tutor to his son (Philopator), he became librarian at
Alexandria ; his obligation to Ptolemy he recognized by the
column which he erected with a graceful epigram inscribed on
it. This is the epigram, with which we are already acquainted
(vol. i, p. 260), relating to the solutions, discovered up to date,
of the problem of the duplication of the cube, and commend-
ing his own method by means of an appliance called fxeaoXafiov,
itself represented in bronze on the column.
Eratosthenes wrote a book with the title IlXaTcoviKos, and,
whether it was a sort of commentary on the Timaeus of
Plato, or a dialogue in which the principal part was played by
Plato, it evidently dealt with the fundamental notions of
mathematics in connexion with Plato's philosophy. It was
naturally one of the important sources of Theon of Smyrna's
work on the mathematical matters which it was necessary for
the student of Plato to know ; and Theon cites the work
twice by name. It seems to have begun with the famous
problem of Delos, telling the story quoted by Theon how the
god required, as a means of stopping a plague, that the altar
PLATONICUS AND ON MEANS 105
there, which was cubical in form, should be doubled in size.
The book evidently contained a disquisition on 'proportion
(dvaXoyia); a quotation by Theon on this subject shows that
Eratosthenes incidentally dealt with the fundamental defini-
tions of geometry and arithmetic. The principles of music
were discussed in the same work.
We have already described Eratosthenes' s solution of the
problem of Delos, and his contribution to the theory of arith-
metic by means of his sieve (kovkivov) for finding successive
prime numbers.
He wrote also an independent work On means. This was in
two Books, and was important enough to be mentioned by
Pappus along with works by Euclid, Aristaeus and Apol-
lonius as forming part of the Treasury of Analysis 1 ; this
proves that it was a systematic geometrical treatise. Another
passage of Pappus speaks of certain loci which Eratosthenes
called 'loci with reference to means' (tottol irpbs fieo-oTrjTas) 2 ;
these were presumably discussed in the treatise in question.
What kind of loci these were is quite uncertain ; Pappus (if it
is not an interpolator who speaks) merely says that these loci
' belong to the aforesaid classes of loci ', but as the classes are
numerous (including ' plane ', ' solid ', ' linear ', ' loci on surfaces ',
&c), we are none the wiser. Tannery conjectured that they
were loci of points such that their distances from three fixed
straight lines furnished a ' mediete^', i.e. loci (straight lines
and conies) which we should represent in trilinear coordinates
by such equations as 2y = x + z, y 2 = xz, y(x + z) = 2xz,
x(x — y) — z(y — z), x(x — y) = y(y — z), the first three equations
representing the arithmetic, geometric and harmonic means,
while the last two represent the ' subcontraries ' to the
harmonic and geometric means respectively. Zeuthen has
a different conjecture. 3 He points out that, if QQ' be the
polar of a given point C with reference to a conic, and GPOP'
be drawn through meeting QQ f in and the conic in P, P f ,
then GO is the harmonic mean to GP, GP' ; the locus of for
all transversals GPP' is then the straight line QQ\ If A, G
are points on PP f such that GA is the arithmetic, and GG the
1 Pappus, vii, p. 636. 24. 2 lb., p. 662. 15 sq.
3 Zeuthen, Die Lehre von den Kegelschnitten im Altertum, 1886, pp.
320, 321.
106 ERATOSTHENES
geometric mean between CP, CP' ', the loci of A, G respectively
are conies. Zeuthen therefore suggests that these loci and
the corresponding loci of the points on CPP' at a distance
from C equal to the subcontraries of the geometric and
harmonic means between CP and CP' are the 'loci with
reference to means ' of Eratosthenes ; the latter two loci are
'linear', i.e. higher curves than conies. Needless to say, we
have no confirmation of this conjecture.
Eratosthenes s measurement of the Earth.
But the most famous scientific achievement of Eratosthenes
was his measurement of the earth. Archimedes mentions, as
we have seen, that some had tried to prove that the circum-
ference of the earth is about 300,000 stades. This was
evidently the measurement based on observations made at
Lysimachia (on the Hellespont) and Syene. It was observed
that, while both these places were on one meridian, the head
of Draco was in the zenith at Lysimachia, and Cancer in the
zenith at Syene ; the arc of the meridian separating the two
in the heavens was taken to be 1/I5th of the complete circle.
^ . The distance between the two towns
was estimated at 20,000 stades, and
accordingly the whole circumference of
the earth was reckoned at 300,000
stades. Eratosthenes improved on this.
He observed (1) that at Syene, at
noon, at the summer solstice, the
sun cast no shadow from an upright
gnomon (this was confirmed by the
observation that a well dug at the
same place was entirely lighted up at
the same time), while (2) at the same moment the gnomon fixed
upright at Alexandria (taken to be on the same meridian with
Syene) cast a shadow corresponding to an angle between the
gnomon and the sun's rays of l/50th of a complete circle or
four right angles. The sun's rays are of course assumed to be
parallel at the two places represented by S and A in the
annexed figure. If oc be the angle made at A by the sun's rays
with the gnomon (DA produced), the angle SO A is also equal to
MEASUREMENT OF THE EARTH 107
a, or l/50th of four right angles. Now the distance from S
to A was known by measurement to be 5,000 stades ; it
followed that the circumference of the earth was 250,000
stades. This is the figure given by Cleomedes, but Theon of
Smyrna and Strabo both give it as 252,000 stades. The
reason of the discrepancy is not known ; it is possible that
Eratosthenes corrected 250,000 to 252,000 for some reason,
perhaps in order to get a figure divisible by 60 and, inci-
dentally, a round number (700) of stades for one degree. If
Pliny is right in saying that Eratosthenes made 40 stades
equal to the Egyptian a\o1vos, then, taking the o-yolvos at
12,000 Royal cubits of 0-525 metres, we get 300 such cubits,
or 157-5 metres, i.e. 516-73 feet, as the length of the stade.
On this basis 252,000 stades works out to 24,662 miles, and
the diameter of the earth to about 7,850 miles, only 50 miles
shorter than the true polar diameter, a surprisingly close
approximation, however much it owes to happy accidents
in the calculation.
We learn from Heron's Dioptra that the measurement of
the earth by Eratosthenes was given in a separate work On
the Measurement of the Earth. According to Galen 1 this work
dealt generally with astronomical or mathematical geography,
treating of ' the size of the equator, the distance of the tropic
and polar circles, the extent of the polar zone, the size and
distance of the sun and moon, total and partial eclipses of
these heavenly bodies, changes in the length of the day
according to the different latitudes and seasons'. Several
details are preserved elsewhere of results obtained by
Eratosthenes, which were doubtless contained in this work.
He is supposed to have estimated the distance between the
tropic circles or twice the obliquity of the ecliptic at 1 l/83rds
of a complete circle or 47° 42' 39"; but from Ptolemy's
language on this subject it is not clear that this estimate was
not Ptolemy's own. What Ptolemy says is that he himself
found the distance between the tropic circles to lie always
between 47° 40' and 47° 45', 'from which we obtain about
(ayeSov) the same ratio as that of Eratosthenes, which
Hipparchus also used. For the distance between the tropics
becomes (or is found to be, yiverai) very nearly 1 1 parts
Galen, Instit. Logica, 12 (p. 26 Kalbfleisch).
108 ERATOSTHENES
out of 83 contained in the whole meridian circle'. 1 The
mean of Ptolemy's estimates, 4 7° 42' 30", is of course nearly
ll/83rds of 360°. It is consistent with Ptolemy's language
to suppose that Eratosthenes adhered to the value of the
obliquity of the ecliptic discovered before Euclid's time,
namely 24°, and Hipparchus does, in his extant Commentary
on the Phaenomena of Aratus and Eudoxus, say that the
summer tropic is ' very nearly 24° north of the equator'.
The Doxographi state that Eratosthenes estimated the
distance of the moon from the earth at 780,000 stades and
the distance of the sun from the earth at 804,000,000 stades
(the versions of Stobaeus and Joannes Lydus admit 4,080,000
as an alternative for the latter figure, but this obviously
cannot be right). Macrobius 2 says that Eratosthenes made
the 'measure' of the sun to be 27 times that of the earth.
It is not certain whether measure means ' solid content ' or
' diameter ' in this case ; the other figures on record make the
former more probable, in which case the diameter of the sun
would be three times that of the earth. Macrobius also tells
us that Eratosthenes's estimates of the distances of the sun
and moon were obtained by means of lunar eclipses.
Another observation by Eratosthenes, namely that at Syene
(which is under the summer tropic) and throughout a circle
round it with a radius of 300 stades the upright gnomon
throws no shadow at noon, was afterwards made use of by
Posidonius in his calculation of the size of the sun. Assuming
that the circle in which the sun apparently moves round the
earth is 10,000 times the size of a circular section of the earth
through its centre, and combining with this hypothesis the
datum just mentioned, Posidonius arrived at 3,000,000 stades
as the diameter of the sun.
Eratosthenes wrote a poem called Hermes containing a good
deal of descriptive astronomy ; only fragments of this have
survived. The work Catasterismi (literally ' placings among
the stars ') which is extant can hardly be genuine in the form
in which it has reached us ; it goes back, however, to a genuine
work by Eratosthenes which apparently bore the same name ;
alternatively it is alluded to as KardXoyoi or by the general
1 Ptolemy, Syntaxis, i. 12, pp. 67. 22-68. 6.
2 Macrobius, In Somn. Scip. i. 20. 9.
ASTRONOMY, ETC. 109
word 'Ao-TpovofjLia (Suidas), which latter word is perhaps a mis-
take for 'Ao-rpoOeo-la corresponding to the title 'AcrrpoOeo-icu
(coSloov found in the manuscripts. The work as we have it
contains the story, mythological and descriptive, of the con-
stellations, &c., under forty-four heads ; there is little or
nothing belonging to astronomy proper.
Eratosthenes is also famous as the first to attempt a scientific
chronology beginning from the siege of Troy; this was the
subject of his Xpovoypa(piai, with which must be connected
the separate 'OXv/imovLKai in several books. Clement of
Alexandria gives a short resumS of the main results of the
former work, and both works were largely used by Apollo-
dorus. Another lost work was on the Octaeteris (or eight-
years' period), which is twice mentioned, by Geminus and
Achilles ; from the latter we learn that Eratosthenes re-
garded the work on the same subject attributed to Eudoxus
as not genuine. His Geographica in three books is mainly
known to us through Suidas's criticism of it. It began with
a history of geography down to his own time ; Eratosthenes
then proceeded to mathematical geography, the spherical form
of the earth, the negligibility in comparison with this of the
unevennesses caused by mountains and valleys, the changes of
features due to floods, earthquakes and the like. It would
appear from Theon of Smyrna's allusions that Eratosthenes
estimated the height of the highest mountain to be 10 stades
or about 1/ 8000th part of the diameter of the earth.
XIV
CONIC SECTIONS. APOLLONIUS OF PERGA
A. HISTORY OF CONICS UP TO APOLLONIUS
Discovery of the conic sections by Menaechmus.
We have seen that Menaechmus solved the problem of the
two mean proportionals (and therefore the duplication of
the cube) by means of conic sections, and that he is credited
with the discovery of the three curves ; for the epigram of
Eratosthenes speaks of ' the triads of Menaechmus ', whereas
of course only two conies, the parabola and the rectangular
hyperbola, actually appear in Menaechmus's solutions. The
question arises, how did Menaechmus come to think of obtain-
ing curves by cutting a cone 1 On this we have no informa-
tion whatever. Democritus had indeed spoken of a section of
a cone parallel and very near to the base, which of course
would be a circle, since the cone would certainly be the right
circular cone. But it is probable enough that the attention
of the Greeks, whose observation nothing escaped, would be
attracted to the shape of a section of a cone or a cylinder by
a plane obliquely inclined to the axis when it occurred, as it
often would, in real life ; the case where the solid was cut
right through, which would show an ellipse, would presum-
ably be noticed first, and some attempt would be made to
investigate the nature and geometrical measure of the elonga-
tion of the figure in relation to the circular sections of the
same solid ; these would in the first instance be most easily
ascertained when the solid was a right cylinder ; it would
then be a natural question to investigate whether the curve
arrived at by cutting the cone had the same property as that
obtained by cutting the cylinder. As we have seen, the
DISCOVERY OF THE CONIC SECTIONS 111
observation that an ellipse can be obtained from a cylinder
as well as a cone is actually made by Euclid in his Phaeno-
mena : 'if, says Euclid, ' a cone or a cylinder be cut by
a plane not parallel to the base, the resulting section is a
section of an acute-angled cone which is similar to a Ovpeos
(shield).' After this would doubtless follow the question
what sort of curves they are which are produced if we
cut a cone by a plane which does not cut through the
cone completely, but is either parallel or not parallel to
a generator of the cone, whether these curves have the
same property with the ellipse and with one another, and,
if not, what exactly are their fundamental properties respec-
tively.
As it is, however, we are only told how the first writers on
conies obtained them in actual practice. We learn on the
authority of Geminus l that the ancients defined a cone as the
surface described by the revolution of a right-angled triangle
about one of the sides containing the right angle, and that
they knew no cones other than right cones. Of these they
distinguished three kinds ; according as the vertical angle of
the cone was less than, equal to, or greater than a right angle,
they called the cone acute-angled, right-angled, or obtuse-
angled, and from each of these kinds of cone they produced
one and only one of the three sections, the section being
always made perpendicular to one of the generating lines of
the cone ; the curves were, on this basis, called ' section of an
acute-angled cone' (= an ellipse), ' section of a right-angled
cone' (= a parabola), and 'section of an obtuse-angled cone '
(= a hyperbola) respectively. These names were still used
by Euclid and Archimedes.
Menaechmuss probable procedure.
Menaechmus's constructions for his curves would presum-
ably be the simplest and the uyost direct that would show the
desired properties, and for the parabola nothing could be
simpler than a section of a right-angled cone by a plane at right
angles to one of its generators. Let OBG (Fig. 1) represent
1 Eutocius, Comm. on Conies of Apollonius.
112
CONIC SECTIONS
a section through the axis OL of a right-angled cone, and
conceive a section through AG (perpendicular to OA) and at
right angles to the plane of the paper.
a/
L
F
Fig. 1.
If P is any point on the curve, and PN perpendicular to
A G, let BG be drawn through N perpendicular to the axis of
the cone. Then P is on the circular section of the cone about
BG as diameter.
Draw AD parallel to BG, and DF, GG parallel to GL meet-
ing AL produced in F, G. Then AD, AF are both bisected
by OL.
If now PN = y, AN = x,
1/= PN 2 = BN.NG.
But B, A, G, G are concyclic, so that
BN.NG=AN.NG
= AN.AF
= AN.2AL.
Therefore f = AN.2AL
= 2AL.x,
and 2AL is the ' parameter ' of the principal ordinates y.
In the case of the hyperbola Menaechmus had to obtain the
MENAECHMUS'S PROCEDURE
113
particular hyperbola which we call rectangular or equilateral,
and also to obtain its property with reference to its asymp-
totes, a considerable advance on what was necessary in the
case of the parabola. Two methods of obtaining the particular
hyperbola were possible, namely (1) to obtain the hyperbola
arising from the section of any obtuse-angled cone by a plane
at right angles to a generator, and then to show how a
rectangular hyperbola can be obtained as a particular case
by finding the vertical angle which the cone must have to
give a rectangular hyperbola when cut in the particular way,
or (2) to obtain the rectangular hyperbola direct by cutting
another kind of cone by a section not necessarily perpen-
dicular to a generator.
(1) Taking the first method, we draw (Fig. 2) a cone with its
vertical angle BOG obtuse. Imagine a section perpendicular
to the plane of the paper and passing through AG which is
perpendicular to OB. Let GA produced meet GO produced in
A\ and complete the same construction as in the case of
the parabola.
Fig. 2.
In this case we have
1523.2
PN 2 = BN.NG = AN.NG.
i
114 CONIC SECTIONS
But, by similar triangles,
NG:AF=NC:AD
= A'N:AA'.
Hence PlY 2 = A N . A'N . ^-,
AA
AN.A'N.
AA f
which is the property of the hyperbola, A A' being what we
call the transverse axis, and 2 AL the parameter of the principal
ordinates.
Now, in order that the hyperbola may be rectangular, we
must have 2 AL : AA f equal to 1. The problem therefore now
is: given a straight line AA\ and AL along A' 'A produced
equal to \ A A\ to find a cone such that L is on its axis and
the section through AL perpendicular to the generator through
A is a rectangular hyperbola with A' A as transverse axis. In
other words, we have to find a point on the straight line
through A perpendicular to AA f such that OX bisects the
angle which is the supplement of the angle A'OA.
This is the case if A'O : OA = A'L : LA = 3:1;
therefore is on the circle which is the locus of all points
such that their distances from the two fixed points A' , A
are in the ratio 3:1. This circle is the circle on K L as
diameter, where A'K-.KA = A'L: LA = 3:1. Draw this
circle, and is then determined as the point in which AO
drawn perpendicular to AA' intersects the circle.
It is to be observed, however, that this deduction of a
particular from a more general case is not usual in early
Greek mathematics ; on the contrary, the particular usually
led to the more general. Notwithstanding, therefore, that the
orthodox method of producing conic sections is said to have
been by cutting the generator of each cone perpendicularly,
I am inclined to think that Menaechmus would get his rect-
angular hyperbola directly, and in an easier way, by means of
a different cone differently cut. Taking the right-angled cone,
already used for obtaining a parabola, we have only to make
a section parallel to the axis (instead of perpendicular to a
generator) to get a rectangular hyperbola.
MENAECHMUS'S PROCEDURE
115
Q
n
A
P v
\
M
N
For, let the right-angled cone HOK (Fig. 3) be cut by a
plane through A'AN parallel
to the axis OM and cutting the
sides of the axial triangle HOK
in A f , A, JV" respectively. Let
P be the point on the curve
for which PN is the principal
ordinate. Draw 00 parallel
to HK. We have at once H,
PN 2 = HN.NK ^
— MK 2 —MN 2
- mn. m±y Fjg g
= CN 2 -CA 2 , since MK = OM, and MN = 0(7= 0^.
This is the property of the rectangular hyperbola having A' A
as axis. To obtain a particular rectangular hyperbola with
axis of given length we have only to choose the cutting plane
so that the intercept A 'A may have the given length.
But Menaechmus had to prove the asymptote-property of
his rectangular hyperbola. As he can hardly be supposed to
have got as far as Apollonius in investigating the relations of
the hyperbola to its asymptotes, it is probably safe to assume
that he obtained the particular property in the simplest way,
i. e. directly from the property of the curve in relation to
its axes.
R
Fig. 4.
If (Fig. 4) CR, CB! be the asymptotes (which are therefore
12
116 CONIC SECTIONS
at right angles) and A' A the axis of a rectangular hyperbola,
P any point on the curve, PN the principal ordinate, draw
PK, PK' perpendicular to the asymptotes respectively. Let
PN produced meet the asymptotes in R, R'.
Now, by the axial property,
CA 2 = CN 2 -PN 2
= RN 2 -PN 2
= RP.PR'
= 2PK. PK', since IPRK is half a right angle ;
therefore PK.PK' = \ CA 2 .
*
Works by Aristaeus and Euclid.
If Menaechmus was really the discoverer of the three conic
sections at a date which we must put at about 360 or 350 B.C.,
the subject must have been developed very rapidly, for by the
end of the century there were two considerable works on
conies in existence, works which, as we learn from Pappus,
were considered worthy of a place, alongside the Conies of
Apollonius, in the Treasury of Analysis. Euclid flourished
about 300 B.C., or perhaps 10 or 20 years earlier; but his
Conies in four books was preceded by a work of Aristaeus
which was still extant in the time of Pappus, who describes it
as ' five books of Solid Loci connected (or continuous, crvve^rj)
with the conies \ Speaking of the relation of Euclid's Conies
in four books to this work, Pappus says (if the passage is
genuine) that Euclid gave credit to Aristaeus for his dis-
coveries in conies and did not attempt to anticipate him or
wish to construct anew the same system. In particular,
Euclid, when dealing with what Apollonius calls the three-
and four-line locus, ' wrote so much about the locus as was
possible by means of the conies of Aristaeus, without claiming
completeness for his demonstrations \* We gather from these
remarks that Euclid's Conies was a compilation and rearrange-
ment of the geometry of the conies so far as known in his
1 Pappus, vii, p. 678. 4.
WORKS BY ARISTAEUS AND EUCLID 117
time, whereas the work of Aristaeus was more specialized and
more original.
' Solid loci ' and ' solid problems \
' Solid loci ' are of course simply conies, but the use of the
title ' Solid loci ' instead of ' conies ' seems to indicate that
the work was in the main devoted to conies regarded as loci.
As we have seen, ' solid loci ' which are conies are distinguished
from ' plane loci ', on the one hand, which are straight lines
and circles, and from ' linear loci ' on the other, which are
curves higher than conies. There is some doubt as to the
real reason why the term ' solid loci ' was applied to the conic
sections. We are told that ' plane ' loci are so called because
they are generated in a plane (but so are some of the higher
curves, such as the quadratrix and the spiral of Archimedes),
and that ' solid loci ' derived their name from the fact that
they arise as sections of solid figures (but so do some higher
curves, e.g. the spiric curves which are sections of the a-irelpa
or tore). But some light is thrown on the subject by the corre-
sponding distinction which Pappus draws between ' plane ',
' solid ' and ' linear ' problems.
'Those problems', he says, 'which can be solved by means
of a straight line and a circumference of a circle may pro-
perly be called plane ; for the lines by means of which such
problems are solved have their origin in a plane. Those,
however, which are solved by using for their discovery one or
more of the sections of the cone have been called solid ; for
their construction requires the use of surfaces of solid figures,
namely those of cones. There remains a third kind of pro-
blem, that which is called linear ; for other lines (curves)
besides those mentioned are assumed for the construction, the
origin of which is more complicated and less natural, as they
are generated from more irregular surfaces and intricate
movements.' *
The true significance of the word ' plane ' as applied to
problems is evidently, not that straight lines and circles have
their origin in a plane, but that the problems in question can
be solved by the ordinary plane methods of transformation of
1 Pappus, iv, p. 270. 5-17.
118 CONIC SECTIONS
areas, manipulation of simple equations between areas and, in
particular, the application of areas ; in other words, plane
problems were those which, if expressed algebraically, depend
on equations of a degree not higher than the second.
Problems, however, soon arose which did not yield to ' plane '
methods. One of the first was that of the duplication of the
cube, which was a problem of geometry in three dimensions or
solid geometry. Consequently, when it was found that this
problem could be solved by means of conies, and that no
higher curves were necessary, it would be natural to speak of
them as 'solid' loci, especially as they were in fact produced
from sections of a solid figure, the cone. The propriety of the
term would be only confirmed when it was found that, just as
the duplication of the cube depended on the solution of a pure
cubic equation, other problems such as the trisection of an
angle, or the cutting of a sphere into two segments bearing
a given ratio to one another, led to an equation between
volumes in one form or another, i. e. a mixed cubic equation,
and that this equation, which was also a solid problem, could
likewise be solved by means of conies.
Aristaeus's Solid Loci.
The Solid Loci of Aristaeus, then, presumably dealt with
loci which proved to be conic sections. In particular, he must
have discussed, however imperfectly, the locus with respect to
three or four lines the synthesis of which Apollonius says that
he found inadequately worked out in Euclid's Conies. The
theorems relating to this locus are enunciated by Pappus in
this way :
' If three straight lines be given in position and from one and
the same point straight lines be drawn to meet the three
straight lines at given angles, and if the ratio of the rectangle
contained by two of the straight lines so drawn to the square
on the remaining one be given, then the point will lie on a
solid locus given in position, that is, on one of the three conic
sections. And if straight lines be so drawn to meet, at given
angles, four straight lines given in position, and the ratio of
the rectangle contained by two of the lines so drawn to the
rectangle contained by the remaining two be given, then in
ARISTAEUS'S SOLID LOCI 119
the same way the point will lie on a conic section given in
position.' x
The reason why Apollonius referred in this connexion to
Euclid and not to Aristaeus was probably that it was Euclid's
work that was on the same lines as his own.
A very large proportion of the standard properties of conies
admit of being stated in the form of locus-theorems ; if a
certain property holds with regard to a certain point, then
that point lies on a conic section. But it may be assumed
that Aristaeus's work was not merely a collection of the
ordinary propositions transformed in this way ; it would deal
with new locus- theorems not implied in the fundamental
definitions and properties of the conies, such as those just
mentioned, the theorems of the three- and four-line locus.
But one (to us) ordinary property, the focus-directrix property,
was, as it seems to me, in all probability included.
Focus-directrix property known to Euclid.
It is remarkable that the directrix does not appear at all in
Apollonius's great treatise on conies. The focal properties of
the central conies are given by Apollonius, but the foci are
obtained in a different way, without any reference to the
directrix; the focus of the parabola does not appear at all.
We may perhaps conclude that neither did Euclid's Conies
contain the focus-directrix property ; for, according to Pappus,
Apollonius based his first four books on Euclid's four books,
while filling them out and adding to them. Yet Pappus gives
the proposition as a lemma to Euclid's Surface-Loci, from
which we cannot but infer that it was assumed in that
treatise without proof. If, then, Euclid did not take it from
his own Conies, what more likely than that it was contained
in Aristaeus's Solid Loci ?
Pappus's enunciation of the theorem is to the effect that the
locus of a point such that its distance from a given point is in
a given ratio to its distance from a fixed straight line is a conic
section, and is an ellipse, a parabola, or a hyperbola according
as the given ratio is less than, equal to, or greater than unity.
1 Pappus, vii, p. 678. 15-24.
120
CONIC SECTIONS
Proof from Pappus.
The proof i in the case where the given ratio is different from
unity is shortly as follows.
Let S be the fixed point, SX the perpendicular from S on
the fixed line. Let P be any point on the locus and PN
■+— ♦
KAN SK,'
A'
XV
K'S
perpendicular to SX, so that SP is to NX in the given
ratio (e);
thus e 2 = (PN 2 + SN 2 ) : NX 2 .
Take K on SX such that
e 2 = SN 2 :NK 2 ;
then, if K f be another point on SN, produced if necessary,
such that NK = NK',
e 2 : 1 = (PN 2 + SN 2 ) : NX 2 = SN 2 : NK 2
= PN 2 :(NX 2 -NK 2 )
= PN 2 : XK . XK'.
The positions of N, K, K' change with the position of P.
If A, A' be the points on which N falls when K, K' coincide
with X respectively, we have
SA.AX = SN: NK = e:l= SN:NK'= SA': A'X.
Therefore SX : SA = SK :SN = (l+e):e,
whence (1 +e) :e = (SX-SK) : (SA -SN)
= XK:AN
FOCUS-DIRECTRIX PROPERTY 121
Similarly it can be shown that
(1 *e):e = XK':A'N.
By multiplication, XK . XK' :AN. A'N = (1 - e 2 ) : e 2 ;
and it follows from above, ex aequali, that
PN 2 :AN.A'N=(l~e 2 ):l,
which is the property of a central conic.
When e < 1, A and A' lie on the same side of 1, while
N lies on A A', and the conic is an ellipse ; when e > 1, A and
A / lie on opposite sides of X, while N lies on A' A produced,
and the conic is a hyperbola.
The case where e = 1 and the curve is a parabola is easy
and need not be reproduced here.
The treatise would doubtless contain other loci of types
similar to that which, as Pappus says, was used for the
trisection of an angle : I refer to the proposition already
quoted (vol. i, p. 243) that, if A, B are the base angles of
a triangle with vertex P, and AB = 2 A A, the locus of P
is a hyperbola with eccentricity 2.
Propositions included in Euclid's Conies.
That Euclid's Conies covered much of the same ground as
the first three Books of Apollonius is clear from the language
of Apollonius himself. Confirmation is forthcoming in the
quotations by Archimedes of propositions (1) 'proved in
the elements of conies ', or (2) assumed without remark as
already known. The former class include the fundamental
ordinate properties of the conies in the following forms :
(1) for the ellipse,
PN 2 : AN. A'N = P'N' 2 : AN'. A'N' = BC 2 :AG 2 ;
(2) for the hyperbola,
PN 2 : AN. A'N = P'N' 2 : AN' .A'N';
(3) for the parabola, PN 2 = p a . AN;
the principal tangent properties of the parabola ;
the property that, if there are two tangents drawn from one
point to any conic section whatever, and two intersecting
122 CONIC SECTIONS
chords drawn parallel to the tangents respectively, the rect-
angles contained by the segments of the chords respectively
are to one another as the squares of the parallel tangents ;
the by no means easy proposition that, if in a parabola the
diameter through P bisects the chord QQ' in V, and QD is
drawn perpendicular to PV, then
where p a is the parameter of the principal ordinates and p is
the parameter of the ordinates to the diameter PV.
Conic sections in Archimedes.
But we must equally regard Euclid's Conies as the source
from which Archimedes took most of the other ordinary
properties of conies which he assumes without proof. Before
summarizing these it will be convenient to refer to Archi-
medes's terminology. We have seen that the axes of an
ellipse are not called axes but diameters, greater and lesser ;
the axis of a parabola is likewise its diameter and the other
diameters are ' lines parallel to the diameter ', although in
a segment of a parabola the diameter bisecting the base is
the ' diameter ' of the segment. The two ' diameters ' (axes)
of an ellipse are conjugate. In the case of the hyperbola the
' diameter ' (axis) is the portion of it within the (single- branch)
hyperbola ; the centre is not called the ' centre ', but the point
in which the ' nearest lines to the section of an obtuse-angled
cone' (the asymptotes) meet; the half of the axis (CA) is
' the line adjacent to the axis ' (of the hyperboloid of revolution
obtained by making the hyperbola revolve about its 'diameter'),
and A' A is double of this line. Similarly GP is the line
' adjacent to the axis ' of a segment of the hyperboloid, and
P'P double of this line. It is clear that Archimedes did not
yet treat the two branches of a hyperbola as forming one
curve ; this was reserved for Apollonius.
The main properties of conies assumed by Archimedes in
addition to those above mentioned may be summarized thus.
Central Conies.
1. The property of the ordinates to any diameter PP\
QV 2 :PV.P / V = Q'V' 2 :PV'.P'V.
CONIC SECTIONS IN ARCHIMEDES 123
In the case of the hyperbola Archimedes does not give
any expression for the constant ratios PN 2 : AN. A'N and
QV 2 :PV .P'V respectively, whence we conclude that he had
no conception of diameters or radii of a hyperbola not meeting
the curve.
2. The straight line drawn from the centre of an ellipse, or
the point of intersection of the asymptotes of a hyperbola,
through the point of contact of any tangent, bisects all chords
parallel to the tangent.
3. In the ellipse the tangents at the extremities of either of two
conjugate diameters are both parallel to the other diameter.
4. If in a hyperbola the tangent at P meets the transverse
axis in T, and PN is the principal ordinate, AN > AT. (It
is not easy to see how this could be proved except by means
of the general property that, if PP f be any diameter of
a hyperbola, Q V the ordinate to it from Q, and QT the tangent
at Q meeting P'P in T, then TP : TP' = PV:P'V.)
5. If a cone, right or oblique, be cut by a plane meeting all
the generators, the section is either a circle or an ellipse.
6. If a line between the asymptotes meets a hyperbola and
is bisected at the point of concourse, it will touch the
hyperbola.
7. If x, y are straight lines drawn, in fixed directions respec-
tively, from a point on a hyperbola to meet the asymptotes,
the rectangle xy is constant.
8. If PN be the principal ordinate of P, a point on an ellipse,
and if NP be produced to meet the auxiliary circle in p, the
ratio 'pN : PN is constant.
9. The criteria of similarity of conies and segments of
conies are assumed in practically the same form as Apollonius
gives them.
The Parabola.
1. The fundamental properties appear in the alternative forms
PN 2 : P'N' 2 = AN: AN\ or PN 2 = p a . AN,
QV 2 :Q'V' 2 = PV:PV, or QV 2 = p.PV.
Archimedes applies the term parameter (a irap av Bvvclvtcu
at oltto t&s rofxds) to the parameter of the principal ordinates
124 CONIC SECTIONS
only : p is simply the line to which the rectangle equal to QV 2
and of width equal to PFis applied.
2. Parallel chords are bisected by one straight line parallel to
the axis, which passes through the point of contact of the
tangent parallel to the chords.
3. If the tangent at Q meet the diameter PV in T, and QV be
the ordinate to the diameter, PV = PT.
By the aid of this proposition a tangent to the parabola can
be drawn (a) at a point on it, (b) parallel to a given chord.
4. Another proposition assumed is equivalent to the property
of the subnormal, NG = \ r p a .
5. If QQ' be a chord of a parabola perpendicular to the axis
and meeting the axis in M, while QVq another chord parallel
to the tangent at P meets the diameter through P in V, and
RHK is the principal ordinate of any point R on the curve
meeting PV in H and the axis in K, then PV :PH > or
= MK : KA ; ' for this is proved ' (On Floating Bodies, II. 6).
Where it was proved we do not know ; the proof is not
altogether easy. 1
6. All parabolas are similar.
As we have seen, Archimedes had to specialize in the
parabola for the purpose of his treatises on the Quadrature
of the Parabola, Conoids and Spheroids, Floating Bodies,
Book II, and Plane Equilibriums, Book II ; consequently he
had to prove for himself a number of special propositions, which
have already been given in their proper places. A few others
are assumed without proof, doubtless as being. easy deductions
from the propositions which he does prove. They refer mainly
to similar parabolic segments so placed that their bases are in
one straight line and have one common extremity.
1. If any three similar and similarly situated parabolic
segments BQ X , BQ 2 , BQ 3 lying along the same straight line
as bases (BQ 1 < BQ 2 < BQ 3 ), and if E be any point on the
tangent at B to one of the segments, and EO a straight line
through E parallel to the axis of one of the segments and
meeting the segments in R%, R 2 , R 1 respectively and BQ 3
in 0, then
R,R 2 : R 2 R, = (Q 2 Q 3 : BQ 3 ) . (BQ, : Q, Q 2 ).
1 See Apollonius of Perga, ed. Heath, p. liv.
CONIC SECTIONS IN ARCHIMEDES 125
2. If two similar parabolic segments with bases BQ 1} - BQ 2 be
placed as in the last proposition, and if BR Y R 2 be any straight
line through B meeting the segments in R 1} R 2 respectively,
BQ 1 :BQ 2 = BR 1 :BR 2 .
These propositions are easily deduced from the theorem
proved in the Quadrature of the Parabola, that, if through E,
a point on the tangent at B, a straight line ERO be drawn
parallel to the axis and meeting the curve in R and any chord
BQ through B in 0, then
ER:RO = BO: OQ.
3. On the strength of these propositions Archimedes assumes
the solution of the problem of placing, between two parabolic
segments similar to one another and placed as in the above
propositions, a straight line of a given length and in a direction
parallel to the diameters of either parabola.
Euclid and Archimedes no doubt adhered to the old method
of regarding the three conies as arising from sections of three
kinds of right circular cones (right-angled, obtuse-angled and
acute-angled) by planes drawn in each case at right angles to
a generator of the cone. Yet neither Euclid nor Archimedes
was unaware that the ' section of an acute-angled cone ', or
ellipse, could be otherwise produced. Euclid actually says in
his Phaenomena that ' if a cone or cylinder (presumably right)
be cut by a plane not parallel to the base, the resulting section
is a section of an acute-angled cone which is similar to
a Ovpeos (shield) '. Archimedes knew that the non-circular
sections even of an oblique circular cone made by planes
cutting all the generators are ellipses ; for he shows us how,
given an ellipse, to draw a cone (in general oblique) of which
it is a section and which has its vertex outside the plane
of the ellipse on any straight line through the centre of the
ellipse in a plane at right angles to the ellipse and passing
through one of its axes, whether the straight line is itself
perpendicular or not perpendicular to the plane of the ellipse ;
drawing a cone in this case of course means finding the circular
sections of the surface generated by a straight line always
passing through the given vertex and all the several points of
the given ellipse. The method of proof would equally serve
126 APOLLONIUS OF PERGA
for the other two conies, the hyperbola and parabola, and we
can scarcely avoid the inference that Archimedes was equally
aware that the parabola and the hyperbola could be found
otherwise than by the old method.
The first, however, to base the theory of conies on the
production of all three in the most general way from any
kind of circular cone, right or oblique, was Apollonius, to
whose work we now come.
B. APOLLONIUS OF PERGA
Hardly anything is known of the life of Apollonius except
that he was born at Perga, in Pamphylia, that he went
when quite young to Alexandria, where he studied with the
successors of Euclid and remained a long time, and that
he flourished (yeyove) in the reign of Ptolemy Euergetes
(247-222 B.C.). Ptolemaeus Chennus mentions an astronomer
of the same name, who was famous during the reign of
Ptolemy Philopator (222-205 B.C.), and it is clear that our
Apollonius is meant. As Apollonius dedicated the fourth and
following Books of his Conies to King Attalus I (241-197 B.C.)
we have a confirmation of his approximate date. He was
probably born about 262 B.C., or 25 years after Archimedes.
We hear of a visit to Pergamum, where he made the acquain-
tance of Eudemus of Pergamum, to whom he dedicated the
first two Books of the Conies in the form in which they have
come down to us ; they were the first two instalments of a
second edition of the work.
The text of the Conies.
The Conies of Apollonius was at once recognized as the
authoritative treatise on the subject, and later writers regu-
larly cited it when quoting propositions in conies. Pappus
wrote a number of lemmas to it ; Serenus wrote a commen-
tary, as also, according to Suidas, did Hypatia. Eutocius
(fl. a.d. 500) prepared an edition of the first four Books and
wrote a commentary on them ; it is evident that he had before
him slightly differing versions of the completed work, and he
may also have had the first unrevised edition which had got
into premature circulation, as Apollonius himself complains in
the Preface to Book I.
THE TEXT OF THE CONICS 127
The edition of Eutocius suffered interpolations which were
probably made in the ninth century when, under the auspices
of Leon, mathematical studies were revived at Constantinople ;
for it was at that date that the uncial manuscripts were
written, from which our best manuscripts, V (= Cod. Vat. gr.
206 of the twelfth to thirteenth century) for the Conies, and
W (= Cod. Vat. gr. 204 of the tenth century) for Eutocius,
were copied.
Only the first four Books survive in Greek ; the eighth
Book is altogether lost, but the three Books V-VII exist in
Arabic. It was Ahmad and al-Hasan, two sons of Muh. b.
Musa b. Shakir, who first contemplated translating the Conies
into Arabic. They were at first deterred by the bad state of
their manuscripts ; but afterwards Ahmad obtained in Syria
a copy of Eutocius's edition of Books I-IV and had them
translated by Hilal b. Abi Hilal al-Himsi (died 883/4).
Books V-VII were translated, also for Ahmad, by Thabit
b. Qurra ( 826-901) from another manuscript. Naslraddm's
recension of this translation of the seven Books, made in 1248,
is represented by two copies in the Bodleian, one of the year
1301 (No. 943) and the other of 1626 containing Books V-VII
only (No. 885).
A Latin translation of Books I-IV was published by
Johannes Baptista Memus at Venice in 1537 ; but the first
important edition was the translation by Commandinus
(Bologna, 1566), which included the lemmas of Pappus and
the commentary of Eutocius, and was the first attempt to
make the book intelligible by means of explanatory notes.
For the Greek text Commandinus used Cod. Marcianus 518
and perhaps also Vat. gr. 205, both of which were copies of V,
but not V itself.
The first published version of Books V-VII was a Latin
translation by Abraham Echellensis and Giacomo Alfonso
Borelli (Florence, 1661) of a reproduction of the Books written
in 983 by Abu 1 Fath al-Isfahanl.
The editio princeps of the Greek text is the monumental
work of Halley (Oxford, 1710). The original intention was
that Gregory should edit the four Books extant in Greek, with
Eutocius's commentary and a Latin translation, and that
Halley should translate Books V-VII from the Arabic into
128 APOLLONIUS OF PERGA
Latin. Gregory, however, died while the work was proceeding,
and Halley then undertook responsibility for the whole. The
Greek manuscripts used were two, one belonging to Savile
and the other lent by D. Baynard ; their whereabouts cannot
apparently now be traced, but they were both copies of Paris,
gr. 2356, which was copied in the sixteenth century from Paris,
gr. 2357 of the sixteenth century, itself a copy of V. For the
three Books in Arabic Halley used the Bodleian MS. 885, but
also consulted (a) a compendium of the three Books by 'Abdel-
melik al-Shirazi (twelfth century), also in the Bodleian (913),
(b) Borelli's edition, and (c) Bodl. 943 above mentioned, by means
of which he revised and corrected his translation when com-
pleted. Halley 's edition is still, so far as I know, the only
available source for Books V-VII, except for the beginning of
Book V (up to Prop. 7) which was edited by L. Nix (Leipzig,
1889).
The Greek text of Books I-IV is now available, with the
commentaries of Eutocius, the fragments of Apollonius, &c,
in the definitive edition of Heiberg (Teubner, 1891-3).
Apollonius's own account of the Conies.
A general account of the contents of the great work which,
according to Geminus, earned for him the title of the ' great
geometer' cannot be better given than in the words of the
writer himself. The prefaces to the several Books contain
interesting historical details, and, like the prefaces of Archi-
medes, state quite plainly and simply in what way the
treatise differs from those of his predecessors, and how much
in it is claimed as original. The strictures of Pappus (or
more probably his interpolator), who accuses him of being a
braggart and unfair towards his predecessors, are evidently
unfounded. The prefaces are quoted by v. Wilamowitz-
Moellendorff as specimens of admirable Greek, showing how
perfect the style of the great mathematicians could be
when they were free from the trammels of mathematical
terminology.
Book I. General Preface.
Apollonius to Eudemus, greeting.
If you are in good health and things are in other respects
as you wish, it is well ; with me too things are moderately
THE CONIGS 129
well. During the time I spent with you at Pergamum
I observed your eagerness to become acquainted with my
work in conies; I am therefore sending you the first book,
which I have corrected, and I will forward the remaining
books when I have finished them to my satisfaction. I dare
say you have not forgotten my telling you that I undertook
the investigation of this subject at the request of Naucrates
the geometer, at the time when he came to Alexandria and
stayed with me, and, when I had worked it out in eight
books, I gave them to him at once, too hurriedly, because he
was on the point of sailing; they had therefore not been
thoroughly revised, indeed I had put down everything just as
it occurred to me, postponing revision till the end. Accord-
ingly I now publish, as opportunities serve from time to time,
instalments of the work as they are corrected. In the mean-
time it has happened that some other persons also, among
those whom I have met, have got the first and second books
before they were corrected ; do not be surprised therefore if
you come across them in a different shape.
Now of the eight books the first four form an elementary
introduction. The first contains the modes of producing the
three sections and the opposite branches (of the hyperbola),
and the fundamental properties subsisting in them, worked
out more fully and generally than in the writings of others.
The second book contains the properties of the diameters and
the axes of the sections as well as the asymptotes, with other
things generally and necessarily used for determining limits
of possibility (Siopio-fioi) ; and what I mean by diameters
and axes respectively you will learn from this book. The
third book contains many remarkable theorems useful for
the syntheses of solid loci and for diorismi ; the most and
prettiest of these theorems are new, and it was their discovery
which made me aware that Euclid did not work out the
synthesis of the locus with respect to three and four lines, but
only a chance portion of it, and that not successfully ; for it
was not possible for the said synthesis to be completed without
the aid of the additional theorems discovered by me. The
fourth book shows in how many ways the sections of cones
can meet one another and the circumference of a circle ; it
contains other things in addition, none of which have been
discussed by earlier writers, namely the questions in how
many points a section of a cone or a circumference of a circle
can meet [a double-branch hyperbola, or two double-branch
hyperbolas can meet one another].
The rest of the books are more by way of surplusage
(7r€piov(TLa(TTLK(OT€pa) : one of them deals somewhat fully with
1523.2 K
130 APOLLONIUS OF PERGA
minima and maxima^ another with equal and similar sections
of cones, another with theorems of the nature of determina-
tions of limits, and the last with determinate conic problems.
But of course, when all of them are published, it will be open
to all who read them to form their own judgement about them,
according to their own individual tastes. Farewell.
The preface to Book II merely says that Apollonius is
sending the second Book to Eudemus by his son Apollonius,
and begs Eudemus to communicate it to earnest students of the
subject, and in particular to Philonides the geometer whom
Apollonius had introduced to Eudemus at Ephesus. There is
no preface to Book III as we have it, although the preface to
Book IV records that it also was sent to Eudemus.
Preface to Book IV.
Apollonius to Attalus, greeting.
Some time ago I expounded and sent to Eudemus of Per-
gamum the first three books of my conies which I have
compiled in eight books, but, as he has passed away, I have
resolved to dedicate the remaining books to you because of
your earnest desire to possess my works. I am sending you
on this occasion the fourth book. It contains a discussion of
the question, in how many points at most it is possible for
sections of cones to meet one another and the circumference
of a circle, on the assumption that they do not coincide
throughout, and further in how many points at most a
section of a cone or the circumference of a circle can meet the
hyperbola with two branches, [or two double-branch hyper-
bolas can meet one another]; and, besides these questions,
the book considers a number of others of a similar kind.
Now the first question Conon expounded to Thrasydaeus, with-
out, however, showing proper mastery of the proofs, and on
this ground Nicoteles of Cyrene, not without reason, fell foul
of him. The second matter has merely been mentioned by
Nicoteles, in connexion with his controversy with Conon,
as one capable of demonstration ; but I have not found it
demonstrated either by Nicoteles himself or by any one else.
The third question and the others akin to it I have not found
so much as noticed by any one. All the matters referred to,
which I have not found anywhere, required for their solution
many and various novel theorems, most of which I have,
as a matter of fact, set out in the first three books, while the
rest are contained in the present book. These theorems are
of considerable use both for the syntheses of problems and for
THE CONICS 131
diorismi. Nicoteles indeed, on account of his controversy
with Conon, will not have it that any use can be made of the
discoveries of Conon for the purpose of diorismi; he is,
however, mistaken in this opinion, for, even if it is possible,
without using them at all, to arrive at results in regard to
limits of possibility, yet they at all events afford a readier
means of observing some things, e.g. that several or so many
solutions are possible, or again that no solution is possible ;
and such foreknowledge secures a satisfactory basis for in-
vestigations, while the theorems in question are again useful
for the analyses of diorismi. And, even apart from such
usefulness, they will be found worthy of acceptance for the
sake of the demonstrations themselves, just as we accept
many other things in mathematics for this reason and for
no other.
The prefaces to Books V-VII now to be given are repro-
duced for Book V from the translation of L. Nix and for
Books VI, VII from that of Halley.
Preface to Book V.
Apollonius to Attalus, greeting.
In this fifth book I have laid down propositions relating to
maximum and minimum straight lines. You must know
that my predecessors and contemporaries have only super-
ficially touched upon the investigation of the shortest lines,
and have only proved what straight lines touch the sections
and. conversely, what properties they have in virtue of which
they are tangents. For my part, 1 have proved these pro-
perties in the first book (without however making any use, in
the proofs, of the doctrine of the shortest lines), inasmuch as
I wished to place them in close connexion with that part
of the subject in which I treat of the production of the three
conic sections, in order to show at the same time that in each
of the three sections countless properties and necessary results
appear, as they do with reference to the original (transverse)
diameter. The propositions in which I discuss the shortest
lines I have separated into classes, and I have dealt with each
individual case by careful demonstration ; I have also con-
nected the investigation of them with the investigation of
the greatest lines above mentioned, because I considered that
those who cultivate this science need them for obtaining
a knowledge of the analysis, and determination of limits of
possibility, of problems as well as for their synthesis : in
addition to which, the subject is one of those which seem
worthy of study for their own sake. Farewell.
k2
132 APOLLONIUS OF PERGA
Preface to Book VI.
Apollonius to Attalus, greeting.
I send you the sixth book of the conies, which embraces
propositions about conic sections and segments of conies equal
and unequal, similar and dissimilar, besides some other matters
left out by those who have preceded me. In particular, you
will find in this book how, in a given right cone, a section can
be cut which is equal to a given section, and how a right cone
can be described similar to a given cone but such as to contain
a given conic section. And these matters in truth I have
treated somewhat more fully and clearly than those who wrote
before my time on these subjects. Farewell.
Preface to Book VII.
Apollonius to Attalus, greeting.
I send to you with this letter the seventh book on conic
sections. In it are contained a large number of new proposi-
tions concerning diameters of sections and the figures described
upon them ; and all these propositions have their uses in many
kinds of problems, especially in the determination of the
limits of their possibility. Several examples of these occur
in the determinate conic problems solved and demonstrated
by me in the eighth book, which is by way of an appendix,
and which I will make a point of sending to you as soon
as possible. Farewell.
Extent of claim to originality.
We gather from these prefaces a very good idea of the
plan followed by Apollonius in the arrangement of the sub-
ject and of the extent to which he claims originality. The
first four Books form, as he says, an elementary introduction,
by which he means an exposition of the elements of conies,
that is, the definitions and the fundamental propositions
which are of the most general use and application ; the term
' elements ' is in fact used with reference to conies in exactly
the same sense as Euclid uses it to describe his great work.
The remaining Books beginning with Book V are devoted to
more specialized investigation of particular parts of the sub-
ject. It is only for a very small portion of the content of the
treatise that Apollonius claims originality ; in the first three
Books the claim is confined to certain propositions bearing on
the ' locus with respect to three or four lines ' ; and in the
fourth Book (on the number of points at which two conies
THE CONICS 133
may intersect, touch, or both) the part which is claimed
as new is the extension to the intersections of the parabola,
ellipse, and circle with the double-branch hyperbola, and of
two double-branch hyperbolas with one another, of the in-
vestigations which had theretofore only taken account of the
single-branch hyperbola. Even in Book V, the most remark-
able of all, Apollonius does not say that normals as ' the shortest
lines ' had not been considered before, but only that they had
been superficially touched upon, doubtless in connexion with
propositions dealing with the tangent properties. He explains
that he found it convenient to treat of the tangent properties,
without any reference to normals, in the first Book in order
to connect them with the chord properties. It is clear, there-
fore, that in treating normals as maxima and minima, and by
themselves, without any reference to tangents, as he does in
Book V, he was making an innovation ; and, in view of the
extent to which the theory of normals as maxima and minima
is developed by him (in 77 propositions), there is no wonder
that he should devote a whole Book to the subject. Apart
from the developments in Books III, IV, V, just mentioned,
and the numerous new propositions in Book VII with the
problems thereon which formed the lost Book VIII, Apollonius
only claims to have treated the whole subject more fully and
generally than his predecessors.
Great generality of treatment from the beginning.
So far from being a braggart and taking undue credit to
himself for the improvements which he made upon his prede-
cessors, Apollonius is, if anything, too modest in his descrip-
tion of his personal contributions to the theory of conic
sections. For the ' more fully and generally ' of his first
preface scarcely conveys an idea of the extreme generality
with which the whole subject is worked out. This character-
istic generality appears at the very outset.
Analysis of the Conies.
Book I.
Apollonius begins by describing a double oblique circular
cone in the most general way. Given a circle and any point
outside the plane of the circle and in general not lying on the
134 APOLLONIUS OF PERGA
straight line through the centre of the circle perpendicular to
its plane, a straight line passing through the point and pro-
duced indefinitely in both directions is made to move, while
always passing through the fixed point, so as to pass succes-
sively through all the points of the circle ; the straight line
thus describes a double cone which is in general oblique or, as
Apollonius calls it, scalene. Then, before proceeding to the
geometry of a cone, Apollonius gives a number of definitions
which, though of course only required for conies, are stated as
applicable to any curve.
1 In any curve,' says Apollonius, ' I give the name diameter to
any straight line which, drawn from the curve, bisects all the
straight lines drawn in the curve (chords) parallel to any
straight line, and I call the extremity of the straight line
(i.e. the diameter) which is at the curve a vertex of the curve
and each of the parallel straight lines (chords) an ordinate
(lit. drawn ordinate- wise, reray/zej/o)? KaTrj-^Oai) to the
diameter/
He then extends these terms to a pair of curves (the primary
reference being to the double-branch hyperbola), giving the
name transverse diameter to any straight line bisecting all the
chords in both curves which are parallel to a given straight
line (this gives two vertices where the diameter meets the
curves respectively), and the name erect diameter (6p6ia) to
any straight line which bisects all straight lines drawn
between one curve and the other which are parallel to any
straight line ; the ordinates to any diameter are again the
parallel straight lines bisected by it. Conjugate diameters in
any curve or pair of curves are straight lines each of which
bisects chords parallel to the other. Axes are the particular
diameters which cut at right angles the parallel chords which
they bisect ; and conjugate axes are related in the same way
as conjugate diameters. Here we have practically our modern
definitions, and there is a great advance on Archimedes's
terminology.
The conies obtained in the most general way from an
oblique cone.
Having described a cone (in general oblique), Apollonius
defines the axis as the straight line drawn from the vertex to
THE C0NIC8, BOOK I 135
the centre of the circular base. After proving that all
sections parallel to the base are also circles, and that there
is another set of circular sections subcontrary to these, he
proceeds to consider sections of the cone drawn in any
manner. Taking any triangle through the axis (the base of
the triangle being consequently a diameter of the circle which
is the base of the cone), he is careful to make his section cut
the base in a straight line perpendicular to the particular
diameter which is the base of the axial triangle. (There is
no loss of generality in this, for, if any section is taken,
without reference to any axial triangle, we have only to
select the particular axial triangle the base of which is that
diameter of the circular base which is
at right angles to the straight line in
which the section of the cone cuts the
base.) Let ABC be any axial triangle,
and let any section whatever cut the
base in a straight line DE at right
angles to EC; if then PM be the in-
tersection of the cutting plane and the
axial triangle, and if QQ / be any chord
in the section parallel to DE, Apollonius
proves that QQ' is bisected by PM. In
other words, PM is a diameter of the section. Apollonius is
careful to explain that,
' if the cone is a right cone, the straight line in the base (DE)
will be at right angles to the common section (PM) of the
cutting plane and the triangle through the axis, but, if the
cone is scalene, it will not in general be at right angles to PM,
but will be at right angles to it only when the plane through
the axis (i.e. the axial triangle) is at right angles to the base
of the cone ' (I. 7).
That is to say, Apollonius works out the properties of the
conies in the most general way with reference to a diameter
which is not one of the principal diameters or axes, but in
general has its ordinates obliquely inclined to it. The axes do
not appear in his exposition till much later, after it has been
shown that each conic has the same property with reference
to any diameter as it has with reference to the original
diameter arising out of the construction ; the axes then appear
136
APOLLONIUS OF PERGA
as particular cases of the new diameter of reference. The
three sections, the parabola, hyperbola, and ellipse are made
in the manner shown in the figures. In each case they pass
through a straight line DE in the plane of the base which
is at right angles to BC, the base of the axial triangle, or
to BG produced. The diameter PM is in the case of the
THE CONIGS, BOOK I 137
parabola parallel to AG \ in the case of the hyperbola it meets
the other half of the double cone in P' ; and in the case of the
ellipse it meets the cone itself again in P f . We draw, in
the cases of the hyperbola and ellipse, AF parallel to PM
to meet BG or BG produced in F.
Apollonius expresses the properties of the three curves by
means of a certain straight line PL drawn at right angles
to PM in the plane of the section.
In the case of the parabola, PL is taken such that
PL:PA = BG 2 : BA . AC;
and in the case of the hyperbola and ellipse such that
PL:PP'=BF.FC:AF 2 .
In the latter two cases we join P'L, and then draw VR
parallel to PL to meet P'L, produced if necessary, in R.
If HK be drawn through V parallel to BG and meeting
AB, A C in H, K respectively, HK is the diameter of the circular
section of the cone made by a plane parallel to the base.
Therefore Q V 2 = HV . VK.
Then (1) for the parabola we have, by parallels and similar
triangles, »
HV:PV=BC:GA,
and VK:PA = BC:BA.
138 APOLLONIUS OF PERGA
Therefore QV 2 :PV.PA = HV.VK:PV.PA
= BC 2 :BA.AC
= PL: PA, by hypothesis,
= PL.PV:PV.PA,
whence QV 2 = PL . PV.
(2) In the case of the hyperbola and ellipse,
HV:PV = BF:FA,
VK:P'V=FC:AF.
Therefore QV 2 : PV. P'V = HV . VK : PV.P'V
= BF.FC:AF 2
= PIj : PP', by hypothesis,
= RV:P'V
= PV. VR.PV.P'V,
whence QV 2 = PV.VR.
Neiv names, ' parabola ', ' ellipse ', ' hyperbola \
Accordingly, in the case of the parabola, the square of the
ordinate (QV 2 ) is equal to the rectangle applied to PL and
with width equal to the abscissa (PV) ;
in the case of the hyperbola the rectangle applied to PL
which is equal to QV 2 and has its width equal to the abscissa
PV overlaps or exceeds (u7r6p/3d\\€i) by the small rectangle LR
which is similar and similarly situated to the rectangle con-
tained by PL, PP' ;
in the case of the ellipse the corresponding rectangle falls
short (e\\ei7r€i) by a rectangle similar and similarly situated
to the rectangle contained by PL, PP'.
Here then we have the properties of the three curves
expressed in the precise language of the Pythagorean applica-
tion of areas, and the curves are named accordingly : parabola
(7rapa/3o\rj) where the rectangle is exactly applied, hyperbola
(v7r€p/3o\r)) where it exceeds, and ellipse (eAAei^i?) where it
falls short. ■
THE CONICS, BOOK I 139
PL is called the latus rectum (opQia) or the parameter of
the ordinates (nap' t)v Bvvavrai at Karayonevcu reTay/Jiei'cos) in
each case. In the case of the central conies, the diameter PP'
is the transverse (fj irXayLa) or transverse diameter', while,
even more commonly, Apollonius speaks of the diameter and
the corresponding parameter together, calling the latter the
latus rectum or erect side (6p6la TrXevpd) and the former
the transverse side of the figure (e?#o?) on, or applied to, the
diameter.
Fundamental properties equivalent to Cartesian equations.
If p is the parameter, and d the corresponding diameter,
the properties of the curves are the equivalent of the Cartesian
equations, referred to the diameter and the tangent at its
extremity as axes (in general oblique),
y 2 = px (the parabola),
y 2 =.px ±--jX 2 (the hyperbola and ellipse respectively).
Thus Apollonius expresses the fundamental property of the
central conies, like that of the parabola, as an equation
between areas, whereas in Archimedes it appears as a
proportion
y 2 : (a 2 + x 2 ) = b 2 : a 2 ,
which, however, is equivalent to the Cartesian equation
referred to axes with the centre as origin. The latter pro-
perty with reference to the original diameter is separately
proved in I. 21, to the effect that QV 2 varies as PV.P'V, as
is really evident from the fact that QV 2 :PV .P'V = PL: PP',
seeing that PL : PP' is constant for any fixed diameter PP'.
Apollonius has a separate proposition (I. 14) to prove that
the opposite branches of a hyperbola have the same diameter
and equal latera recta corresponding thereto. As he was the
first to treat the double-branch hyperbola fully, he generally
discusses the hyperbola (i.e. the single branch) along with
the ellipse, and the opposites, as he calls the double-branch
hyperbola, separately. The properties of the single-branch
hyperbola are, where possible, included in one enunciation
with those of the ellipse and circle, the enunciation beginning,
140 APOLLONIUS OF PERGA
' If in a hyperbola, an ellipse, or the circumference of a circle ' ;
sometimes, however, the double-branch hyperbola and the
ellipse come in one proposition, e.g. in I. 30: 'If in an ellipse
or the opposites (i. e. the double hyperbola) a straight line be
drawn through the centre meeting the curve on both sides of
the centre, it will be bisected at the centre.' The property of
conjugate diameters in an ellipse is proved in relation to
the original diameter of reference and its conjugate in I. 15,
where it is shown that, if DD' is the diameter conjugate to
PP' (i.e. the diameter drawn ordinate- wise to PP'), just as
PP' bisects all chords parallel to DD', so DD' bisects all chords
parallel to PP' ; also, if DL' be drawn at right angles to DD'
and such that DL' . DD' = PP' 2 (or DL' is a third proportional
to DD', PP'), then the ellipse has the same property in rela-
tion to DD' as diameter and DL' as parameter that it has in
relation to PP' as diameter and PL as the corresponding para-
meter. Incidentally it appears that PL . PP' = DD' 2 , or PL is
a third proportional to PP', DD', as indeed is obvious from the
property of the curve QV 2 : PV. PV'= PL : PP' = DD' 2 : PP' 2 .
The next proposition, I. 16, introduces the secondary diameter
of the double-branch hyperbola (i.e. the diameter conjugate to
the transverse diameter of reference), which does not meet the
curve; this diameter is defined as that straight line drawn
through the centre parallel to the ordinates of the transverse
diameter which is bisected at the centre and is of length equal
to the mean proportional between the ' sides of the figure ',
i.e. the transverse diameter PP' and the corresponding para-
meter PL. The centre is defined as the middle point of the
diameter of reference, and it is proved that all other diameters
are bisected at it (I. 30).
Props. 17-19, 22-9, 31-40 are propositions leading up to
and containing the tangent properties. On lines exactly like
those of Eucl. III. 1 6 for the circle, Apollonius proves that, if
a straight line is drawn through the vertex (i. e. the extremity
of the diameter of reference) parallel to the ordinates to the
diameter, it will fall outside the conic, and no other straight
line can fall between the said straight line and the conic ;
therefore the said straight line touches the conic (1.17, 32).
Props. I. 33, 35 contain the property of the tangent at any
point on the parabola, and Props. I. 34, 36 the property of
THE GONICS, BOOK I 141
the tangent at any point of a central conic, in relation
to the original diameter of reference ; if Q is the point of
contact, QV the ordinate to the diameter through P, and
if QT, the tangent at Q, meets the diameter produced in T,
then (1) for the parabola PV = PT, and (2) for the central
conic TP : TP' = PV: VP'. The method of proof is to take a
point T on the diameter produced satisfying the respective
relations, and to prove that, if TQ be joined and produced,
any point on TQ on either side of Q is outside the curve : the
form of proof is by reductio ad absurdum, and in each
case it is again proved that no other straight line can fall
between TQ and the curve. The fundamental property
TP-.TP' = PV-.VP' for the central conic is then used to
prove that GV . GT = GP 2 and QV 2 : CV . VT = p: PP' (or
CD 2 : GP 2 ) and the corresponding properties with reference to
the diameter DD / conjugate to PP' and v, t, the points where
DD' is met by the ordinate to it from Q and by the tangent
at Q respectively (Props. I. 37-40).
Transition to neiv diameter and tangent at its extremity.
An important section of the Book follows (I. 41-50), con-
sisting of propositions leading up to what amounts to a trans-
formation of coordinates from the original diameter and the
tangent at its extremity to any diameter and the tangent at
its extremity ; what Apollonius proves is of course that, if
any other diameter be taken, the ordinate-property of the
conic with reference to that diameter is of the same form as it
is with reference to the original diameter. It is evident that
this is vital to the exposition. The propositions leading up to
the result in I. 50 are not usually given in our text-books of
geometrical conies, but are useful and interesting.
Suppose that the tangent at any point Q meets the diameter
of reference P V in T, and that the tangent at P meets the
diameter through Q in E. Let R be any third point on
the curve; let the ordinate RW to PV meet the diameter
through Q in F, and let RU parallel to the tangent at Q meet
PV in U. Then
(1) in the parabola, the triangle RUW = the parallelogram
EW ' and.
142
APOLLONIUS OF PERGA
/
THE CONIGS, BOOK I 143
(2) in the hyperbola or ellipse, ARUW = the difference
between the triangles GFW and CPE.
(1) In the parabola ARUW: AQTV = RW 2 : QV 2
= PW:PV
= EJEW:njEV.
But, since TV = 2PV, AQTV=CJEV:
therefore A R UW = O EW.
(2) The proof of the proposition with reference to the
central conic depends on a Lemma, proved in I. 4 1 , to the effect
that, if PX, VY be similar parallelograms on CP, GV as bases,
and if VZ be an equiangular parallelogram on Q V as base and
such that, if the ratio of CP to the other side of PX is m, the
ratio of QV to the other side of VZ is m . p / ' PP', then VZ is
equal to the difference between VY and PX. The proof of the
Lemma by Apollonius is difficult, but the truth of it can be
easily seen thus.
By the property of the curve, QV 2 : CV 2 ^CP 2 = p: PP' ;
PP'
therefore CV 2 - CP 2 = ~ . QV 2 .
P
Now LJPX = p. CP 2 /m, where /i is a constant depending
on the angle of the parallelogram.
Similarly
CJVY=a.CV 2 /m, and E3VZ = a ——QV 2 /m.
p r
It follows that DFF- DPI = o VZ.
Taking now the triangles GFW, CPE and R U W in the
ellipse or hyperbola, we see that GFW, CPE are similar, and
RUW has one angle (at W) equal or supplementary to the
angles at P and V in the other two triangles, while we have
QV*:CV.YT = <p.PP' i
whence QV: VT = (p : PP') . (CV: QV),
and, by parallels,
RW: WU=(p: PP') . (CP : PE).
144 APOLLONIUS OF PERGA
Therefore RUW, CPE, CFW are the halves of parallelograms
related as in the lemma ;
therefore A RUW = A CFW - A CPE.
The same property with reference to the diameter secondary
to CPV is proved in I. 45.
It is interesting to note the exact significance of the property
thus proved for the central conic. The proposition, which is
the foundation of Apollonius's method of transformation of
coordinates, amounts to this. If OP, CQ are fixed semi-
diameters and R a variable point, the area of the quadrilateral
GFRU is constant for all positions of R on the conic. Suppose
now that CP, CQ are taken as axes of x and y respectively.
If we draw RX parallel to CQ to meet CP and RY parallel to
CP to meet CQ, the proposition asserts that (subject to the
proper convention as to sign)
ARYF+CJCXRY+ARXU = (const).
But since RX, RY, RF, RU are in fixed directions,
ARYF varies as RY 2 or x 2 , C3CXRY as RX . RY or xy,
and ARXU as RX 2 or if.
Hence, if x, y are the coordinates of R,
ocx 2 + fixy + yy 2 — A,
which is the Cartesian equation of the conic referred to the
centre as origin and any two diameters as axes.
The properties so obtained are next used to prove that,
if UR meets the curve again in R f and the diameter through
Q in M, then RR' is bisected at M. (I. 46-8).
Taking (1) the case of the parabola, we have,
ARUW=EJEW,
and AR'UW'=CJEW'.
By subtraction, (RWW'R) = CJF'W,
whence ARFM = AR'F'M,
and, since the triangles are similar, RM — R'M.
The same result is easily obtained for the central conic.
It follows that EQ produced in the case of the parabola,
THE COmCS, BOOK I 145
or CQ in the case of the central conic, bisects all chords as
RR' parallel to the tangent at Q. Consequently EQ and CQ
are diameters of the respective conies.
In order to refer the conic to the new diameter and the
corresponding ordinates, we have only to determine the para-
meter of these ordinates and to show that the property of the
conic with reference to the new parameter and diameter is in
the same form as that originally found.
The propositions I. 49, 50 do this, and show that the new
parameter is in all the cases p', where (if is the point of
intersection of the tangents at P and Q)
0Q:QE = p':2QT.
(l) In the case of the parabola, we have TP = PV = EQ,
whence AEOQ = APOT.
Add to each the figure POQF'W ';
therefore QTW'F' = CJEW ' = AR'UW,
whence, subtracting MUW'F' from both, we have
AR'MF' = ejQU.
Therefore R'M . M F' = 2QT. QM.
But R'M : MF' = 0Q:QE = p':2 QT, by hypothesis ;
therefore R'M 2 : R'M . MF' = p' . QM :2QT. QM.
And R'M. MF' = 2QT . QM, from above ;
therefore R'M 2 = p' . QM,
which is the desired property. 1
1 The proposition that, in the case of the parabola, if p be the para-
meter of the ordinates to the diameter through Q, then (see the first figure
on p. 142)
0Q:QE = p:2QT
has an interesting application ; for it enables us to prove the proposition,
assumed without proof by Archimedes (but not easy to prove otherwise),
that, if in a parabola the diameter through P bisects the chord QQ' in V,
and QD is drawn perpendicular to PV, then
QV*:QD* = p:p a ,
1523.2 L
146
APOLLONIUS OF PERGA
(2) In the case of the central conic, we have
AR'UW = ACF'W - AGPE.
(Apollonius here assumes what he does not proye till III. 1,
namely that AGPE = ACQT. This is proved thus.
We have GV: GT = GV 2 : CP 2 ; (I. 37, 39.)
therefore AGQV: ACQT = ACQV: AGPE,
so that ACQT = AGPE.)
Therefore AR'UW' = ACF'W ' * ACQT,
and it is easy to prove that in all cases
AR'MF'=QTUM.
Therefore R'M . MF' = QM(QT + MU).
Let QL be drawn at right angles to CQ and equal to p'.
Join Q'L and draw MK parallel to QL to meet Q'L in K.
Draw CH parallel to Q'L to meet QL in H and MK in N.
Now RfM: MF' = OQ:QE
— QL : 2 QT, by hypothesis,
= QH:QT.
But QT : MU = CQ : CM = QH: MN,
so that (QH + MN) :{QT + MU) = QH:QT
= R'MiMF', from above.
where i) a is the parameter of the principal ordinates and p the para-
meter of the ordinates to the diameter
PV.
If the tangent at the vertex A meets
VP produced in E, and PT, the tangent
at P, in 0, the proposition of Apollonius
proves that
0P:PE = p:2PT.
But
therefore
Thus
OP
_ i
PT 2 =p.PE
= p.AN.
QV 2 : QD' 1 = PT 2 : PN 2 , by similar triangles,
= p . AN:p u . AN
THE CONICS, BOOK I 147
It follows that
QM(QH+MN) : QM(QT + MU) = R'M* :R'M . MF' ;
but, from above, QM(QT+MU) = R'M . MF'\
therefore R'M* = QAI(QH+ MN)
= QM.MK,
which is the desired property.
In the case of the hyperbola, the same property is true for
the opposite branch.
These important propositions show that the ordinate property
of the three conies is of the same form whatever diameter is
taken as the diameter of reference. It is therefore a matter
of indifference to which particular diameter and ordinates the
conic is referred. This is stated by Apollonius in a summary
which follows I. 50.
First appearance of principal axes.
The axes appear for the first time in the propositions next
following (I. 52-8), where Apollonius shows how to construct
each of the conies, given in each case (1) a diameter, (2) the
length of the corresponding parameter, and (3) the inclination
of the ordinates to the diameter. In each case Apollonius
first assumes the angle between the ordinates and the diameter
to be a right angle ; then he reduces the case where the angle
is oblique to the case where it is right by his method of trans-
formation of coordinates; i.e. from the given diameter and
parameter he finds the axis of the conic and the length of the
corresponding parameter, and he then constructs the conic as
in the first case where the ordinates are at right angles to the
diameter. Here then w T e have a case of the proof of existence
by means of construction. The conic is in each case con-
structed by finding the cone of which the given conic is a
section. The problem of finding the axis of a parabola and
the centre and the axes of a central conic when the conic (and
not merely the elements, as here) is given comes later (in II.
44-7), where it is also proved (II. 48) that no central conic
can have more than two axes.
L 2
148 APOLLONIUS OF PERGA
It has been my object, by means of the above detailed
account of Book I, to show not merely what results are
obtained by Apollonius, but the way in which he went to
work ; and it will have been realized how entirely scientific
and general the method is. When the foundation is thus laid,
and the fundamental properties established, Apollonius is able
to develop the rest of the subject on lines more similar to
those followed in our text-books. My description of the rest
of the work can therefore for the most part be confined to a
summary of the contents.
Book II begins with a section devoted to the properties of
the asymptotes. They are constructed in II. 1 in this way.
Beginning, as usual, with any diameter of reference and the
corresponding parameter and inclination of ordinates, Apol-
lonius draws at P the vertex (the extremity of the diameter)
a tangent to the hyperbola and sets off along it lengths PL, PL'
on either side of P such that PL 2 =PL' 2 =±p . PP' [ = GD%
where p is the parameter. He then proves that CL, GU pro-
duced will not meet the curve in any finite point and are there-
fore asymptotes. II. 2 proves further that no straight line
through G within the angle between the asymptotes can itself
be an asymptote. II. 3 proves that the intercept made by the
asymptotes on the tangent at any point P is bisected at P, and
that the square on each half of the intercept is equal to one-
fourth of the ' figure ' corresponding to the diameter through
P (i.e. one-fourth of the rectangle contained by the 'erect'
side, the latus rectum or parameter corresponding to the
diameter, and the diameter itself) ; this property is used as a
means of drawing a hyperbola when the asymptotes and one
point on the curve are given (II. 4). II. 5-7 are propositions
about a tangent at the extremity of a diameter being parallel
to the chords bisected by it. Apollonius returns to the
asymptotes in II. 8, and II. 8-14 give the other ordinary
properties with reference to the asymptotes (II. 9 is a con-
verse of II. 3), the equality of the intercepts between the
asymptotes and the curve of any chord (II. 8), the equality of
the rectangle contained by the distances between either point
in which the chord meets the curve and the points of inter-
section with the asymptotes to the square on the parallel
semi-diameter (II. 10), the latter property with reference to
THE CONIGS, BOOK II 149
the portions of the asymptotes which include between them
a branch of the conjugate hyperbola (II. 11), the constancy of
the rectangle contained by the straight lines drawn from any
point of the curve in fixed directions to meet the asymptotes
(equivalent to the Cartesian equation with reference to the
asymptotes, xy = const.) (II. 12), and the fact that the curve
and the asymptotes proceed to infinity and approach con-
tinually nearer to one another, so that the distance separating
them can be made smaller than any given length (II. 14). II. 15
proves that the two opposite branches of a hyperbola have the
same asymptotes and II. 16 proves for the chord connecting
points on two branches the property of II. 8. II. 1 7 shows that
'conjugate opposites' (two conjugate double-branch hyper-
bolas) have the same asymptotes. Propositions follow about
conjugate hyperbolas; any tangent tcPthe conjugate hyper-
bola will meet both branches of the original hyperbola
and will be bisected at the point of contact (II. 19); if Q be
any point on a hyperbola, and GE parallel to the tangent
at Q meets the conjugate hyperbola in E, the tangent at
E will be parallel to GQ and GQ, GE will be conjugate
diameters (II. 20), while the tangents at Q, E will meet on one
of the asymptotes (II. 21) ; if a chord Qq in one branch of
a hyperbola meet the asymptotes in R, r and the conjugate
hyperbola in Q', q', then Q'Q.Qq' = 2 CD 2 (II. 23). Of the
rest of the propositions in this part of the Book the following
may be mentioned : if TQ, TQ' are two tangents to a conic
and V is the middle point of QQ', TV is a diameter (II. 29,
30, 38) ; if tQ, tQ' be tangents to opposite branches of a hyper-
bola, RR' the chord through t parallel to QQ', v the middle
point of QQ', then vR, vR' are tangents to the hyperbola
(II. 40) ; in a conic, or a circle, or in conjugate hyperbolas, if
two chords not passing through the centre intersect, they do not
bisect each other (II. 26, 41, 42). II. 44-7 show how to find
a diameter of a conic and the centre of a central conic, the
axis of a parabola and the axes of a central conic. The Book
concludes with problems of drawing tangents to conies in
certain ways, through any point on or outside the curve
(II. 49), making with the axis an angle equal to a given acute
angle (II. 50), making a given angle with the diameter through
the point of contact (II. 51, 53) ; II. 52 contains a Siopio-pos for
150 APOLLONIUS OF PERGA
the last problem, proving that, if the tangent to an ellipse at
any point P meets the major axis in T, the angle GPT is not
greater than the angle ABA', where B is one extremity of the
minor axis.
Book III begins with a series of propositions about the
equality of certain areas, propositions of the same kind as, and
easily derived from, the propositions (I. 41-50) by means of
which, as already shown, the transformation of coordinates is
effected. We have first the proposition that, if the tangents
at any points P, Q of a conic meet in 0, and if they meet
the diameters through Q, P respectively in E, T, then
AOPT = A0QE (III. 1, 4) ; and, if P, Q be points on adjacent
branches of conjugate hyperbolas, AGPE = ACQT (III. 13.).
With the same notation, if R be any other point on the conic,
and if we draw BU parallel to the tangent at Q meeting the
diameter through P^ in U and the diameter through Q in M,
and RW parallel to the tangent at P meeting QT in H and
the diameters through Q, P in F, W, then AHQF = quadri-
lateral HTUR (III. 2. 6) ; this is proved at once from the fact
that ABMF= quadrilateral QTUM (see I. 49, 50, or pp. 145-6
above) by subtracting or adding the area HRMQ on each
side. Next take any other point B', and draw B'U', F'H'B'W
in the same way as before ; it is then proved that, if BU, R'W
meet in I and B'U', R W in J, the quadrilaterals F'IBF, IUU'R'
are equal, and also the quadrilaterals FJB'F', JU'TJR (III. 3,
7, 9, 10). The proof varies according to the actual positions
of the points in the figures.
In Figs. 1, 2 AHFQ = quadrilateral HTUR,
AH'F'Q = H'TU'R'.
• By subtraction, FHH'F'= IUU'R + (IB);
whence, if IE be added or subtracted, F'IRF = IUU'R',
and again, if I J be added to both, FJR'F' = JU'UR.
In Fig. 3 AR'U' W = A CF'W - A CQT,
so that ACQT= CU'R'F'.
THE CONICS, BOOK III
151
E F 1 F
T U l)' P
Fig. 1.
Fig. 2.
Fig. 3.
152 APOLLONIUS OF PERGA
Adding the quadrilateral CF'H'T, we have
AH'F'Q = H'TU'R',
and similarly AHFQ = HTUR.
By subtraction, F'H'HF= H'TU'R' -HTUR.
Adding H'IRH to each side, we have
F'IRF == IUU'R'.
If each of these quadrilaterals is subtracted from //,
FJR'F' = JU'UR.
The corresponding results are proved in III. 5, 11, 12, 14
for the case where the ordinates through RR' are drawn to
a secondary diameter, and in III. 15 for the case where P, Q
are on the original hyperbola and R, R' on the conjugate
hyperbola.
The importance of these propositions lies in the fact that
they are immediately used to prove the well-known theorems
about the rectangles contained by the segments of intersecting
chords and the harmonic properties of the pole and polar.
The former question is dealt with in III. 16-23, which give
a great variety of particular cases. We will give the proof
of one case, to the effect that, if OP, OQ be two tangents
to any conic and Rr, R'r' be any two chords parallel to
them respectively and intersecting in J, an internal or external
point,
then R J . Jr : R f J . Jr' = OP 2 : OQ 2 = (const.).
We have
RJ. Jr = RW 2 ^JW 2 , and RW 2 : JW 2 = ARUW : AJU'W;
therefore
RJ.Jr: RW 2 = (RW 2 - JW 2 ) : RW 2 = JU'UR : ARUW.
But RW 2 : OP 2 = ARUW: AOPT;
therefore, ex aequali, RJ.Jr: OP 2 = JU'UR : A OPT.
THE CONICS, BOOK III 153
Similarly R'M ' 2 : JM' 2 = AR'F'M' : A JFM',
whence R'J . JV' : R'M ' 2 = P/P'P' : A R'F'M'.
But iTilf' 2 : OQ 2 = AR'F'M' : A OQE ;
therefore, ea; aequali, RJ . Jr' : OQ 2 = FJRF' : A OQE.
It follows, since PJE'P' = JU'UR, and AOP2 7 = AOQE,
that iU . JV : OP 2 = E'J" . Jr' : OQ 2 ,
or JBJT . Jr : EV . JV = OP 2 : OQ 2 .
If we had taken chords Ri\, P'r/ parallel respectively to
OQ, OP and intersecting in I, an internal or external point,
we should have in like manner
RI . Ir t : RI . Ir( = OQ 2 : OP 2 .
As a particular case, if PP r be a diameter, and Rr, Rr' be
chords parallel respectively to the tangent at P and the
diameter PP' and intersecting in /, then (as is separately
proved)
RI.Ir:RI.Ir' = p:PP'.
The corresponding results are proved in the cases where certain
of the points lie on the conjugate hyperbola.
The six following propositions about the segments of inter-
secting chords (III. 24-9) refer to two chords in conjugate
hyperbolas or in an ellipse drawn parallel respectively to two
conjugate diameters PP', DD' ', and the results in modern form
are perhaps worth quoting. If Rr, Rr' be two chords so
drawn and intersecting in 0, then
(a) in the conjugate hyperbolas
RO.Or RO . Or' _
CP l ± CD 2 ~ 2 '
and (RO 2 + Or 2 ) : (RO 2 + Or' 2 ) = CP 2 : CD 2 ;
(b) in the ellipse
R0 2 + Or 2 R0 2 + 0r' 2 _
OP 2 ' + ~CW~ " _ 4 *
154
APOLLONIUS OF PERGA
The general propositions containing the harmonic properties
of the pole and polar of a conic are III. 37-40, which prove
that in any conic, if TQ, Tq be tangents, and if Qq the chord
of contact be bisected in V, then
(1) if any straight line through T meet the conic in R', R and
Qq in I, then (Fig. 1) RT : TB! = RI : IR' ;
t u
(2) if any straight line through Fmeet the conic in R, R' and
the parallel through T to Qq in 0, then (Fig. 2)
R0 : OR = RV: VR.
Fig. 2.
The above figures represent theorem (1) for the parabola and
theorem (2) for the ellipse.
4
THE CONICS, BOOK III 155
To prove (1) we have
R'l 2 : IW-H'Q 2 : QH 2 = AH'F'Q : AHFQ = H'TU'R' : HTUR
(III. 2, 3, &c).
Also R'T 2 : TR 2 = R'U' 2 : UR 2 = AR'U'W : A220TT,
and jRT a : Ti? 2 = TW 2 : TW 2 = ATH'W : A TWIT,
so that R'T 2 :TR 2 = ATH'W' - AE'CHF: ATi^TF- A-RETTF
= H'TU'R':HTUR
= R'l 2 : i7? 2 , from above.
To prove (2) we have
RV 2 : 7iT 2 = RU 2 : R'U' 2 = ARUW: AR'U'W,
and also
= HQ 2 : QH' 2 = AHFQ : AH'F'Q = HTUR * : H'TU'R',
so that
RV 2 : VR' 2 = HTUR + ARUWiH'TU'R' + AR'U'W
= ATHWiATHW
= TF 2 : TIP 2
= RO 2 : OR' 2 .
Props. III. 30-6 deal separately with the particular cases
in which (a) the transversal is parallel to an asymptote of the
hyperbola or (6) the chord of contact is parallel to an asymp-
tote, i.e. where one of the tangents is an asymptote, which is
the tangent at infinity.
Next we have propositions about intercepts made by two
tangents on a third : If the tangents at three points of a
parabola form a triangle, all three tangents will be cut by the
points of contact in the same proportion (III. 41) ; if the tan-
gents at the extremities of a diameter PP' of a central conic
are cut in r, r' by any other tangent, Pr . P'r' = CD 2 (III. 42) ;
if the tangents at P, Q to a hyperbola meet the asymptotes in
* Where a quadrilateral, as HTUR in the figure, is a cross-quadri-
lateral, the area is of course the difference between the two triangles
which it forms, as HTW ^ RUW.
156 APOLLONIUS Ot PERGA
L, 1/ and M, M' respectively, then L'M, LM' are both parallel
to PQ (III. 44).
The first of these propositions asserts that, if the tangents at
three points P, Q, R of a parabola form a triangle pqr, then
Pr :rq = tQ: Qp = qp :pR.
From this property it is easy to deduce the Cartesian
equation of a parabola referred to two fixed tangents as
coordinate axes. Taking qR, qP as fixed coordinate axes, we
find the locus of Q thus. Let x, y be the coordinates of Q.
Then, if qp = x Y , qr = y v qR — h, qP — k, we have
s „ rQ == VvzV „ k -Vi = x i
x x -x ~ Qp y 2/1 h-x x '
From these equations we derive
x x — hx, y* — ky ;
also, since — = ^ x a we have f- — = 1.
x 2/i-2/ x i 2/i
By substituting for x 1} y x the values V(hx), V(ky) we
obtain
©'+©'-■•
The focal properties of central conies are proved in
III. 45-52 without any reference to the directrix ; there is
no mention of the focus of a parabola. The foci are called
' the points arising out of the application ' (ra e/c rrjs irapa-
fio\r)s ytuo/xeua arj/ieTa), the meaning being that 8, S' are taken
on the axis AA' such that AS.SA' = AS'.S'A' = \p a .AA'
or CB 2 , that is, in the phraseology of application of areas,
a rectangle is applied to A A' as base equal to one-fourth
part of the ' figure ', and in the case of the hyperbola ex-
ceeding, but in the case of the ellipse falling short, by a
square figure. The foci being thus found, it is proved that,
if the tangents At, A'r' at the extremities of the axis are met
by the tangent at any point P in r, v' respectively, rr' subtends
a right angle at S, S', and the angles rr'S, A'r'S' are equal, as
also are the angles rV/S", ArS (III. 45, 46). It is next shown
that, if be the intersection of r>S r/ , r'S, then OP is perpen-
dicular to the tangent at P (III. 47). These propositions are
THE CO NWS, BOOK III 157
used to prove that the focal distances of P make equal angles
with the tangent at P (III. 48). In III. 49-52 follow the
other ordinary properties, that, if SY be perpendicular to
the tangent at P, the locus of Y is the circle on A A' as
diameter, that the lines from G drawn parallel to the focal
distances to meet the tangent at P are equal to CA, and that
the sum or difference of the focal distances of any point is
equal to A A'.
The last propositions of Book III are of use with reference
to the locus with respect to three or four lines. They are as
follows.
1. If PP' be a diameter of a central conic, and if PQ, P'Q
drawn to any other point Q of the conic meet the tangents at
P', P in R' y R respectively, then PR . P'R' = 4 CD 2 (III. 53).
2. If TQ, TQ' be two tangents to a conic, V the middle point
of QQ', P the point of contact of the tangent parallel to QQ',
and R any other point on the conic, let Qr parallel to TQ'
meet Q'R in r, and Q'r parallel to TQ meet QR in r' ; then
Qr . QY : QQ' 2 = (PV 2 : PT 2 ) . (TQ . TQ': QV 2 ). (Ill 54, 56.)
3. If the tangents are tangents to opposite branches of a
hyperbola and meet in t, and if R, r, r' are taken as before,
while tq is half the chord through t parallel to QQ', then
Qr . QY : QQ' 2 = tQ . tQ' : tq 2 . (III. 55.)
The second of these propositions leads at once to the three-
line locus, and from this we easily obtain the Cartesian
equation to a conic with reference to two fixed tangents as
axes, where the lengths of the tangents are h, k, viz.
(M-')'=»©*-
Book IV is on the whole dull, and need not be noticed at
length. Props. 1-23 prove the converse of the propositions in
Book III about the harmonic properties of the pole and polar
for a large number of particular cases. One of the proposi-
tions (IV. 9) gives a method of drawing two tangents to
a conic from an external point T. Draw any two straight
lines through T cutting the conic in Q, Q' and in R, R' respec-
158 APOLLONIUS OF PERGA
tively. Take on QQ' and 0' on RR' so that TQ', TR' are
harmonically divided. The intersections of 00' produced with
the conic give the two points of contact required.
The remainder of the Book (IV. 24-57) deals with intersecting
conies, and the number of points in which, in particular cases,
they can intersect or touch. IV. 24 proves that no two conies
can meet in such a way that part of one of them is common
to both, while the rest is not. The rest of the propositions
can be divided into five groups, three of which can be brought
under one general enunciation. Group I consists of particular
cases depending on the more elementary considerations affect-
ing conies: e.g. two conies having their concavities in oppo-
site directions will not meet in more than two points (IV. 35);
if a conic meet one branch of a hyperbola, it will not meet
the other branch in more points than two (IV. 37); a conic
touching one branch of a hyperbola with its concave side
will not meet the opposite branch (IV. 39). IV. 36, 41, 42, 45,
54 belong to Jbhis group. Group II contains propositions
(IV. 25, 38, 43, 44, 46, 55) showing that no two conies
(including in the term the double-branch hyperbola) can
intersect in more than four points. Group III (IV. 26, 47, 48,
49, 50, 56) are particular cases of the proposition that two
conies which touch at one point cannot intersect at more than
two other points. Group IV (IV. 27, 28, 29, 40, 51, 52, 53, 57)
are cases of the proposition that no two conies which touch
each other at two points can intersect at any other point.
Group V consists of propositions about double contact. A
parabola cannot touch another parabola in more points than
one (IV. 30); this follows from the property TP = PV. A
parabola, if it fall outside a hyperbola, cannot have double
contact with it (IV. 31); it is shown that for the hyperbola
PV>PT, while for the parabola P'V = P'T] therefore the
hyperbola would fall outside the parabola, which is impossible.
A parabola cannot have internal double contact with an ellipse
or circle (IV. 32). A hyperbola cannot have double contact
with another hyperbola having the same centre (IV. 33) ;
proved by means of OV . CT = GP 2 . If an ellipse have double
contact with an ellipse or a circle, the chord of contact will
pass through the centre (IV. 34).
Book V is of an entirely different order, indeed it is the
THE CONICS, BOOKS IV-V 159
most remarkable of the extant Books. It deals with normals
to conies regarded as maximum and minimum straight lines
drawn from particular points to the curve. Included in it are
a series of propositions which, though worked out by the
purest geometrical methods, actually lead immediately to the
determination of the evolute of each of the three conies ; that
is to say, the Cartesian equations to the evolutes can be easily
deduced from the results obtained by Apollonius. There can
be no doubt that the Book is almost wholly original, and it is
a veritable geometrical tour de force.
Apollonius in this Book considers various points and classes
of points with reference to the maximum or minimum straight
lines which it is possible to draw from them to the conies,
i. e. as the feet of normals to the curve. He begins naturally
with points on the axis, and he takes first the point E where
AE measured along the axis from the vertex A is \p, p being
the principal parameter. The first three propositions prove
generally and for certain particular cases that, if in an ellipse
or a hyperbola AM be drawn at right angles to AA' and equal
to J p, and if CM meet the ordinate PN of any point P of the
curve in H, then PN 2 = 2 (quadrilateral MANH) ; this is a
lemma used in the proofs of later propositions, V. 5, 6, &c.
Next, in V. 4, 5, 6, he proves that, if AE = \p, then AE is the
minimum straight line from E to the curve, and if P be any
other point on it, PE increases as P moves farther away from
A on either side ; he proves in fact that, if PN be the ordinate
from P,
(1) in the case of the parabola PE 9 - = AE 2 + AN 2 ,
(2) in the case of the hyperbola or ellipse
PE 2 = A& + AN* • AA A r, P ,
AA
where of course p = BB' 2 /AA\ and therefore (AA / ±p) / A A'
is equivalent to what we call e 2 , the square of the eccentricity.
It is also proved that EA' is the maximum, straight line from
E to the curve. It is next proved that, if be any point on
the axis between A and E, OA is the minimum straight line
from to the curve and, if P is any other point on the curve,
OP increases as P moves farther from A (V. 7).
160
APOLLONIUS OF PERGA
Next Apollonius takes points G on the axis at a distance
from A greater than ^p, an( ^ ne proves that the minimum
straight line from G to the curve (i.e. the normal) is GP,
where P is such a point that
(1) in the case of the parabola NG = \p ;
(2) in the case of the central conic NG : GN = p. A A' ;
and, if P' is any other point on the conic, P'G increases as P f
moves away from P on either side ; this is proved by show-
ing that
( 1 ) for the parabola P'G 2 = PG 2 + NN' 2 ;
(2) for the central conic P'G 2 = PG 2 + NN /2 . AA . j; V '
.A XL
L^R
J P
As these propositions contain the fundamental properties of
the subnormals, it is worth while to reproduce Apollonius's
proofs.
(1) In the parabola, if G be any point on the axis such that
AG > %p, measure GN towards A equal to \p. Let PN be
the ordinate through N, P / any other point on the curve.
Then shall PG be the minimum ^line from G to the curve, &c.
THE COFICS, BOOK V 161
We have P'N' 2 = p . AN' = 2 NG . AN' ;
and NG 2 = NN' 2 + NG 2 ±2NG. NN',
according to the position of N'.
Therefore P'G 2 = 2NG.AN+NG 2 + NN' 2
= PN 2 + NG 2 + NN' 2
= PG 2 + NN' 2 ;
and the proposition is proved.
(2) In the case of the central conic, take G on the axis such
that AG > \p, and measure GN towards A such that
NG:GN = p:AA / .
Draw the ordinate PN through N, and also the ordinate P'N'
from any other point P'.
We have first to prove the lemma (V. 1, 2, 3) that, if AM be
drawn perpendicular to A A' and equal to \p, and if CM,
produced if necessary, meet PN in H, then
PN 2 = 2 (quadrilateral MANE).
This is easy, for, if AL(= 2AM) be the parameter, and A'L
meet PN in R, then, by the property of the curve,
PN 2 = AN.NR
= AN(NH + AM)
= 2 (quadrilateral MANH).
Let GH, produced if necessary, meet P'N' in H'. From H
draw HI perpendicular to P'H' .
Now, since, by hypothesis, NG : GN — p:AA'
= AM:AC
= HN:NC\
NH = NG, whence also H'N' = N'G.
Therefore NG 2 = 2AHNG, N'G 2 = 2 A H'N' G.
And PN 2 = 2(MANH);
therefore PG 2 = NG 2 + PN 2 = 2 (AMHG).
1523.2 M
162 APOLLONIUS OF PERGA
Similarly, if CM meets P'N' in K,
P'G 2 = N'G 2 + P'N' 2
= 2 AH'F'G + 2{AMKN')
= 2(AMHG) + 2AHH'K.
Therefore, by subtraction,
P'G 2 -PG 2 = 2AHH'K
= HI.(H'I±IK)
= HI. (HI ± IK)
CA+AM
= HP
CA
- AW 2 AA '±P -
~ ' AA' '
which proves the proposition.
If be any point on PG, OP is the minimum straight line
from to the curve, and 0P / increases as P' moves away from
P on either side; this is proved in V. 12. (Since P f G > PG,
Z GPP' > Z GP f P ; therefore, a fortiori, Z OPP' > Z OP'P,
and OP' > OP.)
Apollonius next proves the corresponding propositions with
reference to points on the minor axis of an ellipse. If p' be
the parameter of the ordinates to the minor axis, p '=AA' 2 /BB' ',
or i/= OA 2 /GB. If now E' be so taken that BE'=ip',
then BE' is the maximum straight line from E' to the curve
and, if P be any other point on it, E'P diminishes as P moves
farther from B on either side, and E'B' is the minimum
straight line from E' to the curve. It is, in fact, proved that
E'B 2 - E'P 2 = Bn 2 . P ~ f/ 6 > where Bn is the abscissa of P
(V. 16-18). If be any point on the minor axis such that
BO > BE', then OB is the maximum straight line from to
the curve, &c. (V. 19).
If g be a point on the minor axis such that Bg > BG, but
Bg < f p\ and if Gn be measured towards B so that
Cn : ng = BB' : p',
then n is the foot of the ordinates of two points P such that
Pg is the maximum straight line from g to the curve. Also,
THE CONICS, BOOK V 163
if P' be any other point on it, P'g diminishes as P / moves
farther from P on either side to B or B\ and
p 2 J* 2 '2 /"££' , 2 tM 2 -(7i? 2
p g 2__p g 2 = nn 2 J____ Qr ^2 __
If be any point on P# produced beyond the minor axis, PO
is the maximum straight line from to the same part of the
ellipse for which Pg is a maximum, i.e. the semi-ellipse BPB\
&c. (V. 20-2).
In V. 23 it is proved that, if g is on the minor axis, and gP
a maximum straight line to the curve, and if Pg meets A A'
in G, then GP is the minimum straight line from G to the
curve ; this is proved by similar triangles. Only one normal
can be drawn from any one point on a conic (V. 24-6). The
normal at any point P of a conic, whether regarded as a
minimum straight line from G on the major axis or (in the
case of the ellipse) as a maximum straight line from g on the
minor axis, is perpendicular to the tangent at P (V. 27-30);
in general (1) if be any point within a conic, and OP be
a maximum or a minimum straight line from to the conic,
the straight line through P perpendicular to PO touches the
conic, and (2) if 0' be any point on OP produced outside the
conic, O'P is the minimum straight line from 0' to the conic,
&c. (V. 31-4).
Number of normals from a point.
We now come to propositions about two or more normals
meeting at a point. If the normal at P meet the axis of
a parabola or the axis A A' of a hyperbola or ellipse in G, the
angle PGA increases as P or G moves farther away from A,
but in the case of the hyperbola the angle will always be less
than the complement of half the angle between the asymptotes.
Two normals at points on the same side of A A' will meet on
the opposite side of that axis ; and two normals at points on
the same quadrant of an ellipse as i5 will meet at a point
within the angle AGB f (V. 35-40). In a parabola or an
ellipse any normal PG will meet the curve again; in the
hyperbola, (1) if A A' be not greater than p, no normal can
meet the curve at a second point on the same branch, but
M 2
164 APOLLONIUS OF PERGA
(2) if A A' > p, some normals will meet the same branch again
and others not (V. 41-3).
If P 1 G v P 2 G 2 be normals at points on one side of the axis of
a conic meeting in 0, and if be joined to any other point P
on the conic (it being further supposed in the case of the
ellipse that all three lines 0P 1} 0P 2 , OP cut the same half of
the axis), then
(1) OP cannot be a normal to the curve;
(2) if OP meet the axis in K, and PG be the normal at P, AG
is less or greater than AK according as P does or does not lie
between P x and P 2 .
From this proposition it is proved that (1) three normals at
points on one quadrant of an ellipse cannot meet at one point,
and (2) four normals at points on one semi-ellipse bounded by
the major axis cannot meet at one point (V. 44-8).
In any conic, if M be any point on the axis such that AM
is not greater than J^>, and if be any point on the double
ordinate through M, then no straight line drawn to any point
on the curve on the other side of the axis from and meeting
the axis between A and M can be a normal (V. 49, 50).
Propositions leading immediately to the determination
of the evolute of a conic.
These great propositions are V. 51, 52, to the following
effect :
If AM measured along the axis be greater than \p (but in
the case of the ellipse less than AG), and if MO be drawn per-
pendicular to the axis, then a certain length (y, say) can be
assigned such that
(a) if OM > y, no normal can be drawn through which cuts
the axis ; but, if OP be any straight line drawn to the curve
cutting the axis in K, NK < NG, where PN is the ordinate
and PG the normal at P ;
(b) if OM = y, only one normal can be so drawn through 0,
and, if OP be any other straight line drawn to the curve and
cutt ing the axis in K, NK < NG, as before ;
(c) if 0M<y, two normals can be so drawn through 0, and, if
OP be any other straight line drawn to the curve, NK is
THE CONICS, BOOK V 165
greater or less than NG according as OP is or is not inter-
mediate between the two normals (V. 51, 52).
The proofs are of course long and complicated. The length
y is determined in this way :
(1) In the case of the parabola, measure MH towards the
vertex equal to \p, and divide A H atiV^so that HN X = 2N X A.
The length y is then taken such that
y:P 1 N 1 = N 1 H:HM,
where Pji^ is the ordinate passing through JV^ ;
(2) In the case of the hyperbola and ellipse, we have
AM>\p, so that G A \AM<AA'\p\ therefore, if H be taken
on AM such that OH: HM = AA':p, H will fall between A
and M.
Take two mean proportionals GN X , CI between GA and GH,
and let P X N X be the ordinate through N x .
The length y is then taken such that
y : P x #i = (CM : MH) . (H^ : Nfi).
In the case (b), where OM = y, is the point of intersection
of consecutive normals, i. e. is the centre of curvature at the
point P; and, by considering the coordinates of with reference
to two coordinate axes, we can derive the Cartesian equations
of the evolutes. E.g. (1) in the case of the parabola let the
coordinate axes be the axis and the tangent at the vertex.
Then AM = x, OM — y. Let p = 4 a ; then
HM=2a, N l H-%(x-2a), and AN 1 =±(x-2a).
But y 2 : P^ 2 = N X H 2 : HM 2 , by hypothesis,
or y 2 :^a.AN x = F^ila 2 ;
therefore ay 2 = AN X . N^ 2 ,
= -^-rj (x — 2a) ,
or 27 ay 2 = 4(a? — 2 a) 3 .
(2) In the case of the hyperbola or ellipse we naturally take
GA, GB as axes of x and y. The work is here rather more
complicated, but there is no difficulty in obtaining, as the
locus of 0, the curve
(axf + (fo/) f = (a 2 ±b 2 )$.
166 APOLLONIUS OF PERGA
The propositions V. 53, 54 are particular cases of the pre-
ceding propositions.
Construction of normals.
The next section of the Book (V. 55-63) relates to the con-
struction of normals through various points according to their
position within or without the conic and in relation to the
axes. It is proved that one normal can be drawn through any
internal point and through any external point which is not
on the axis through the vertex A. In particular, if is any
point below the axis AA f of an ellipse, and OM is perpen-
dicular to A A', then, if AM>AC, one normal can always be
drawn through cutting the axis between A and C, but never
more than one such normal (V. 55-7). The points on the
curve at which the straight lines through are normals are
determined as the intersections of the conic with a certain
rectangular hyperbola. The procedure
of Apollonius is equivalent to the fol-
lowing analytical method. Let A M be
the axis of a conic, PGO one of the
normals which passes through the given
point 0, PN the ordinate at P ; and let
OM be drawn perpendicular to the axis.
Take as axes of coordinates the axes in the central conic and,
in the case of the parabola, the axis and the tangent at the
vertex.
If then (x, y) be the coordinates of P and (x Y , y x ) those of
we have y NG
~V\ x^ — x — NG
Therefore (1) for the parabola
y _ \p
— j—j
Vi x i ^ — ~2 P
or ^/-(*i-^)2/-2/i-§P = °; (0
(2) in the ellipse or hyperbola
/ _ b 2 \ b 2
x v\ 1 + ^)- x iy±^-yi x = ' ( 2 )
The intersections of these rectangular hyperbolas respec-
THE CONICS, BOOKS V, VI 167
tively with the conies give the points at which the normals
passing through are normals.
Pappus criticizes the use of the rectangular hyperbola in
the case of the parabola as an unnecessary resort to a ' solid
locus ' ; the meaning evidently is that the same points of
intersection can be got by means of a certain circle taking
the place of the rectangular hyperbola. We can, in fact, from
the equation (1) above combined with y 2 = px, obtain the
circle
(x 2 + y 2 ) - (x 1 + ^p)x-iy 1 y = 0.
The Book concludes with other propositions about maxima
and minima. In particular V. 68-71 compare the lengths of
tangents TQ, TQ f , where Q is nearer to the axis than Q\
V. 72, 74 compare the lengths of two normals from a point
from which only two can be drawn and the lengths of other
straight lines from to the curve ; V. 75-7 compare the
lengths of three normals to an ellipse drawn from a point
below the major axis, in relation to the lengths of other
straight lines from to the curve.
Book VI is of much less interest. The first part (VI. 1-27)
relates to equal (i.e. congruent) or similar conies and segments
of conies ; it is naturally preceded by some definitions includ-
ing those of ' equal ' and ' similar ' as applied to conies and
segments of conies. Conies are said to be similar if, the same
number of ordinates being drawn to the axis at proportional
distances from the vertices, all the ordinates are respectively
proportional to the corresponding abscissae. The definition of
similar segments is the same with diameter substituted for
axis, and with the additional condition that the angles
between the base and diameter in each are equal. Two
parabolas are equal if the ordinates to a diameter in each are
inclined to the respective diameters at equal angles and the
corresponding parameters are equal ; two ellipses or hyper-
bolas are equal if the ordinates to a diameter in each are
equally inclined to the respective diameters and the diameters
as well as the corresponding parameters are equal (VI. 1. 2).
Hyperbolas or ellipses are similar when the 'figure' on a
diameter of one is similar (instead of equal) to the ' figure ' on
a diameter of the other, and the ordinates to the diameters in
168 APOLLONIUS OF PERGA
each make equal angles with them ; all parabolas are similar
(VI. 11, 12,13). No conic of one of the three kinds (para-
bolas, hyperbolas or ellipses) can be equal or similar to a conic
of either of the other two kinds (VI. 3, 14, 15). Let QPQ',
qpq' be two segments of similar conies in which QQ', qq' are
the bases and PV, pv are the diameters bisecting them ; then,
if PT, pt be the tangents at P, p and meet the axes at T, t at
equal angles, and if P V : PT = pv : pt, the segments are similar
and similarly situated, and conversely (VI. 17, 18). If two
ordinates be drawn to the axes of two parabolas, or the major or
conjugate axes of two similar central conies, as PN, P'N' and
pn, p'n' respectively, such that the ratios AN: an and AN': an'
are each equal to the ratio of the respective latera recta, the
segments PP f , pp' will be similar ; also PP' will not be similar
to any segment in the other conic cut off by two ordinates
other than pn, p'n' , and conversely (VI. 21, 22). If any cone
be cut by two parallel planes making hyperbolic or elliptic
sections, the sections will be similar but not equal (VI. 26, 27).
The remainder of the Book consists of problems of con-
struction; we are shown how in a given right cone to find
a parabolic, hyperbolic or elliptic section equal to a given
parabola, hyperbola or ellipse, subject in the case of the
hyperbola to a certain Siopio-fios or condition of possibility
(VI. 28-30); also how to find a right cone similar to a given
cone and containing a given parabola, hyperbola or ellipse as
a section of it, subject again in the case of the hyperbola to
a certain Siopio-fjios (VI. 31-3). These problems recall the
somewhat similar problems in I. 51-9.
Book VII begins with three propositions giving expressions
for AP 2 <{ = AN 2 -\-PN 2 ) in the same form as those for PN 2 in
the statement of the ordinary property. In the parabola AH
is measured along the axis produced (i. e. in the opposite direc-
tion to AN) and of length equal to the latus' rectum, and it is
proved that, for any point P, AP 2 = AN.NH (VII. 1). In
the case of the central conies A A' is divided at H, internally
for the hyperbola and externally for the ellipse (AH being the
segment adjacent to A) so that AH : A' H — p : AA' , where p
is the parameter corresponding to AA' , or p = BB' 2 / AA', and
it is proved that
AP 2 :AN.NH= A A': A'H
THE CONICS, BOOKS VI, VII
169
The same is true if A A' is the minor axis of an ellipse and p
the corresponding parameter (VII. 2, 3).
If AA' be divided at H' as well as H (internally for the
hyperbola and externally for the ellipse) so that i^is adjacent
to A and H' to A', and if A'H: AH = AH' : A'H' = A A' :p,
the lines AH, A'H' (corresponding to p in the proportion) are
called by Apollonius homologues, and he makes considerable
use of the auxiliary points H, H' in later propositions from
VII. 6 onwards. Meantime he proves two more propositions,
which, like VII. 1-3, are by way of lemmas. First, if CD be
the semi-diameter parallel to the tangent at P to a central
conic, and if the tangent meet the axis A A' in T, then
PT 2 : CD 2 = NT: CN. (VII. 4.)
Draw AE, TF at right angles to C A to meet CP, and let AE
meet PT in 0. Then, if p' be the parameter of the ordinates
to CP, we have
ip':PT=OP:PE (I. 49,50.)
= PT:PF,
or *y .PF=PT\
Therefore PT 2 : CD 2 = \p' . PF:£p'. CP
= PF: CP
= NT:GN.
170 APOLLONIUS OF PERGA
Secondly, Apollonius proves that, if PN be a principal
ordinate in a parabola, p the principal parameter, p' the
parameter of the ordinates to the diameter through P, then
p' = p + 4Al¥ (VII. 5); this is proved by means of the same
property as VII. 4, namely \p' :PT ' - OP : PE.
Much use is made in the remainder of the Book of two
points Q and M y where AQ is drawn parallel to the conjugate
diameter CD to meet the curve in Q, and M is the foot of
the principal ordinate at Q ; since the diameter GP bisects
both A A! and QA, it follows that A'Q is parallel to GP.
Many ratios between functions of PP', DD' are expressed in
terms of AM, A'M, MH, MH', AH, A'H,&c. The first pro-
positions of the Book proper (VII. 6, 7) prove, for instance,
that PP' 2 : DD' 2 = ME': MH.
For PT 2 : CD 2 = NT:GN = AM: A'M, by similar triangles.
Also GP 2 :PT 2 = A'Q 2 :AQ 2 .
Therefore, ex aequali,
GP 2 : GD 2 = (AM : A'M) x (A'Q 2 : AQ 2 )
= (AM: A'M) x (A'Q 2 : A'M. MH')
x (A'M.MH': AM. MH) x (AM.MH : AQ 2 )
= (AM: A'M) x (AA': AH') x (A'M: AM)
x (MH':MH) x (A'H:AA'), by aid of VII. 2, 3.
Therefore PP' 2 : DD' 2 = MH' : MH.
Next (VII. 8, 9, 10, 11) the following relations are proved,
namely
(\)AA' 2 :(PP' + DD'f = A'H.MH':{MH'+V(MH.MH')} 2 ,
(2) AA' 2 : PP' .DD' = A'H : V(MH.MH'),
(3) A A' 2 : (PP' 2 + DD' 2 ) = A'H : MH± MH'.
The steps by which these results are obtained are as follows.
First, A A' 2 : PP' 2 = A' H : MH' (oc)
= A'H.MH':MH' 2 .
(This is proved thus :
AA' 2 :PP' 2 =GA 2 :GP 2
= GJST.CT:CP 2
= A'M. A' A : A'Q 2 .
THE CONICS, BOOK VII 171
But A'Q*:A'M.MH'= AA'.AH' (VII. 2, 3)
= AA':A'H
= A'M. AA': A'M. A'H,
so that, alternately,
A'M. AA': A'Q 2 = A'M. A'H : A'M . ME '
= A'H:MH'.)
Next, PP' 2 : DD' 2 = MH': MH, as above, (0)
= MH' 2 :MH.MH',
whence PP': DD' = ikf^T r : </(il/# . Jlfff ')> (y)
and PP' 2 :(PP' + DD'f = MH' 2 :{MH' + V(MH .MH')} 2 ;
(1) above follows from this relation and (a) &£ aequali;
(2) follows from (a) and (y) e# aequali, and (3) from (a)
and (0).
We now obtain immediately the important proposition that
PP' 2 + DD' 2 is constant, whatever be the position of P on an
ellipse or hyperbola (the upper sign referring to the ellipse),
and is equal to AA' 2 + BB' 2 (VII. 12, 13, 29, 30).
For AA' 2 : BB' 2 = AA':p = A'H:AH = A'H: A'H,
by construction ;
therefore A A' 2 :AA' 2 + BB' 2 = A'H: HH' ;
also, from (oc) above,
AA' 2 :PP' 2 = A'H:MH f ;
and, by means of (/?),
pp/2 . (pp/2 ± j)^ 2 ) _ jjf #/ . MH > ± ^ #
= MH':HH'.
Ex aequali, from the last two relations, we have
A A' 2 : (PP' 2 + DD' 2 ) = A'H: HH'
■= A A' 2 : A A' 2 + BB' 2 , from above,
whence PP' 2 + DD' 2 = AA' 2 + BB' 2 .
172 APOLLONIUS OF PERGA
A number of other ratios are expressed in terms of the
straight lines terminating at A, A', H, H', M, M' as follows
(VII. 14-20).
In the ellipse A A'* : PP' 2 * DD' 2 = A'H : 2 CM,
and in the hyperbola or ellipse (if p be the parameter of the
ord mates to PP')
AA' 2 :p 2 = A'H.MH': ME 2 ,
A A' 2 : (PP' + pf = A'H . MH' : (MH + MH') 2 ,
AA' 2 :PP'.p = A'H:MH,
and AA' 2 : (PP' 2 ±p 2 ) = A'H . MH':(MH' 2 + MH 2 ).
Apollonius is now in a position, by means of all these
relations, resting on the use of the auxiliary points H, H', M,
to compare different functions of any conjugate diameters
with the same functions of the axes, and to show how the
former vary (by way of increase or diminution) as P moves
away from A. The following is a list of the functions com-
pared, where for brevity I shall use a, b to represent A A' ', BB';
a', V to represent PP', DD' ; and p, p' to represent the para-
meters of the ordinates to AA', PP' respectively.
In a hyperbola, according as a > or < b, a' > or < b', and the
ratio a':b' decreases or increases as P moves from A on
either side; also, if a = b, a' — b' (VII. 21-3); in an ellipse
a:b > a':b', and the latter ratio diminishes as P moves from
A to B (VII. 24).
In a hyperbola or ellipse a + b<a' + b', and a'-f^/in the
hyperbola increases continually as P moves farther from A,
but in the ellipse increases till a', b' take the position of the
equal conjugate diameters when it is a maximum (VII.
25, 26).
In a hyperbola in which a, b are unequal, or in an ellipse,
a^b>a'^b', and a'^b' diminishes as P moves away from A,
in the hyperbola continually, and in the ellipse till a', b' are
the equal conjugate diameters (VII. 27).
ab < a'b', and a'b' increases as P moves away from A, in the
hyperbola continually, and in the ellipse till a', b' coincide with
the equal conjugate diameters (VII. 28).
VII. 31 is the important proposition that, if PP' , DD' are
THE GONIG& BOOK VII
173
conjugate diameters in an ellipse or conjugate hyperbolas, and
if the tangents at their extremities form the parallelogram
LL'MM', then
the parallelogram LL'MM' — rect. AA' . BB' .
The proof is interesting. Let the tangents at P, D respec-
tively meet the major or transverse axis in T, T'.
Now (by VII. 4) PT 2 : GD 2 = NT:GN\
therefore 2 A GPT : 2 A T'DC = NT : CN.
But 2AGPT:(GL) = PT:CD,
= OP : DT' , by similar triangles,
= (CL):2AT'DG.
That is, (OX) is a mean proportional between 2AGPT and
2ArDC.
Therefore, since >/(NT . OiV) is a mean proportional between
iVT and CUV,
174 APOLLONIUS OF PERGA
2AGPT: (GL) = V(CN. NT) : GN
r a
= PN.^:GN (1.37,39)
= PN.GT:GT.GN .^
CA
= 2AGPT.CA.GB;
therefore (GL) = GA . GB.
The remaining propositions of the Book trace the variations
of different functions of the conjugate diameters, distinguishing
the maximum values, &c. The functions treated are the
following :
p', the parameter of the ordinates to PP' in the hyperbola,
according as A A f is (1) not less than^j, the parameter corre-
sponding to A A', (2) less than p but not less than -|p, (3) less
than \p (VII. 33-5).
PP f ^p\ as compared with AA' ^p> in the hyperbola (VII. 36)
or the ellipse (VII. 37).
PP'+p' „ „ AA f +p in the hyperbola (VII.
38-40) or the ellipse (VII. 41).
PP'.p'* „ „ AA'.p in the hyperbola (VII. 42)
or the ellipse (VII. 43).
PP' 2 +p' 2 „ „ AA' 2 +p 2 in the hyperbola, accord-
ing as (1) A A' is not less than
p, or (2) ^4J/<p,but A A' 2 not
less than \(AA f *pf, or (3)
AA' 2 <\(AA r ~p) 2 (VII.44-6).
PP /2 +p /2 „ „ AA' 2 +p 2 in the ellipse, according
as A A' 2 is not greater, or is
greater, than (AA'+p) 2 (VII.
47, 48).
PP" 2 <^p' 2 „ „ AA' 2 ^p 2 in the hyperbola, accord-
ing as AA' > or < p (VII.
49, 50).
PP' 2 - p' 2 „ „ A A' 2 - p 2 or BB' 2 ^p b 2 in the ellipse,
according as PP f > or < p f
(VII. 51).
THE GO NIGS, BOOK VII 175
As we have said, 'Book VIII is lost. The nature of its
contents can only be conjectured from Apollonius's own
remark that it contained determinate conic problems for
which Book VII was useful, particularly in determining
limits of possibility. Unfortunately, the lemmas of Pappus
do not enable us to form any clearer idea. But it is probable
enough that the Book contained a number of problems having
for their object the finding of conjugate diameters in a given
conic such that certain functions of their lengths have given
values. It was on this assumption that Halley attempted
a restoration of the Book.
If it be thought that the above account of the Gonics is
disproportionately long for a work of this kind, it must be
remembered that the treatise is a great classic which deserves
to be more known than it is. What militates against its
being read in its original form is the great extent of the
exposition (it contains 387 separate propositions), due partly
to the Greek habit of proving particular cases of a general
proposition separately from the proposition itself, but more to
the cumbrousness of the enunciations of complicated proposi-
tions in general terms (without the help of letters to denote
particular points) and to the elaborateness of the Euclidean
form, to which Apollonius adheres throughout.
+
Other works by Apollonius.
Pappus mentions and gives a short indication of the con-
tents of six other works of Apollonius which formed part of the
Treasury of Analysis. 1 Three of these should be mentioned
in close connexion with the Conies.
(a) On the Gutting-off of a Ratio (Xoyou a7roTOfirj),
two Books.
This work alone of the six mentioned has survived, and
that only in the Arabic ; it was published in a Latin trans-
lation by Edmund Halley in 1706. It deals with the general
problem, ' Given two straight lines, parallel to one another or
intersecting, and a fixed point on each line, to draw through
1 Pappus, vii, pp. 640-8, 660-72.
176 APOLLONIUS OF PERGA
a given point a straight line which shall tut off segments from
each line (measured from the fixed points) bearing a given
ratio to one another! Thus, let A, B be fixed points on the
two given straight lines A C, BK, and let be the given
point. It is required to draw through a straight line
cutting the given straight lines in points M, N respectively
such that AM is to BN in a given ratio. The two Books of
the treatise discussed the various . possible cases of this pro-
blem which arise according to the relative positions of the
given straight lines and points, and also the necessary condi-
tions and limits of possibility in cases where a solution is not
always possible. The first Book begins by supposing the
given lines to be parallel, and discusses the different cases
which arise ; Apollonius then passes to the cases in which the
straight lines intersect, but one of the given points, A or B, is
at the intersection of the two lines. Book II proceeds to the
general case shown in the above figure, and first proves that
the general case can be reduced to the case in Book I where
one of the given points, A or B, is at the intersection of the
two lines. The reduction is easy. For join OB meeting AG
in B', and draw B'N' parallel to BN to meet OM in N'. Then
the ratio B'N' : BN, being equal to the ratio OB' : OB, is con-
stant. Since, therefore, BN: A M is a given ratio, the ratio
B'N' : AM is also given.
Apollonius proceeds in all cases by the orthodox method of
analysis and synthesis. Suppose the problem solved and
OMN drawn through in such a way that B'N : AM is a
given ratio = A, say.
ON THE CUTTTNG-OFF OF A RATIO 177
Draw OG parallel to BN or B'N' to meet AM in G. Take
D on AM such that OC:AB =X = B'N' : J.lf,
Then AM : ^D = .S'iV' : 00
= B'M:CM;
therefore MB:AB = B'G : CM,
or CM .MB = AB . BV, a given rectangle.
Hence the problem is reduced to one of applying to GB a
rectangle (GM . MB) equal to a given rectangle (AB . B'G) but
falling short by a square figure. In the case as drawn, what-
ever be the value of A, the solution is always possible because
the given rectangle AB . GB' is always less than CA . AB, and
therefore always less than J- GB 2 ; one of the positions of
M falls between A and B because GM . MB<GA . AB.
The proposition III. 41 of the Conies about the intercepts
made on two tangents to a parabola by a third tangent
(pp. 155-6 above) suggests an obvious application of our pro-
blem. We had, with the notation of that proposition,
Pr : rq =.rQ : Qp = qp :pR.
Suppose that the two tangents qP, qR are given as fixed
tangents with their points of contact P, R. Then we can
draw another tangent if we can draw a straight line
intersecting qP,qR in such a way that Pr:rq=qp:pR or
Pq : qr = qR :pR, i. e. qr \pR = Pq : qR (a constant ratio) ;
i.e. we have to draw a straight line such that the intercept by
it on qP measured from q has a given ratio to the intercept
by it on qR measured from R. This is a particular case of
our problem to which, as a matter of fact, Apollonius devotes
special attention. In the annexed figure the letters have the
same meaning as before, and N'M has to be drawn through
such that B'N' : AM = A. In this case there are limits to
1623.2 N
178 APOLLONIUS OF PERGA
the value of X in order that the solution may be possible.
Apollonius begins by stating the limiting case, saying that we
obtain a solution in a special manner in the case where M is
the middle point of CD, so that the rectangle CM . AID or
CB' . AD has its maximum value.
The corresponding limiting value of X is determined by
finding the corresponding position of D or M.
We have B'C :MD = CM: AD, as before,
= B'M:MA;
whence, since MD = CM,
B'C:B'M = CM:MA
= B'M:B'A,
so that B'M 2 = B'C.B'A.
Thus M is found and therefore D also.
According, therefore, as X is less or greater than the par-
ticular value of OC: AD thus determined, Apollonius finds no
solution or two solutions.
Further, we have
AD = B'A + B'C- (B'D + B'C)
= B'A + B'C-2B'M
= B'A + B'C- 2 VB'A . B'C.
If then we refer the various points to a system of co-
ordinates in which B'A, B'N' are the axes of x and y, and if
we denote by (x, y) and the length B'A by h,
X = 00/ AD = y/(h + x-2Vhx).
If we suppose Apollonius to have used these results for the
parabola, he cannot have failed to observe that the limiting
case described is that in which is on the parabola, while
N'OM is the tangent at ; for, as above,
B'M : B'A = B'C:B'M = N'O : N'M, by parallels,
so that B'A, N'M are divided at M, respectively in the same
proportion.
ON THE CUTTING-OFF OF A RATIO 179
Further, if we put for A the ratio between the lengths of the
two fixed tangents, then if h, k be those lengths,
k y
h h + x-2\/hx
which can easily be reduced to
©*+©-'
the equation of the parabola referred to the two fixed tangents
as axes.
•(/?) On the cutting-off of an area (\coptov ccttoto/jltj),
two Books.
This work, also in two Books, dealt with a similar problem,
with the difference that the intercepts on the given straight
lines measured from the given points are required, not to
have a given ratio, but to contain a given rectangle. Halley
included an attempted restoration of this work in his edition
of the De sectione rationis.
The general case can here again be reduced to the more
special one in which one of the fixed points is at the inter-
section of the two given straight lines. Using the same
figure as before, but with D taking the position shown by (D)
in the figure, we take that point such that
OC . AD — the given rectangle.
We have then to draw ON'M through such that
B'N' .AM=OC.AD,
or B'N':OC=AD:AM.
But, by parallels, B''N' : OC = B'M: CM;
therefore AM :CM=AD: B'M
= MD:B'C,
so that B'M .MD = AD. B'C.
Hence, as before, the problem is reduced to an application
of a rectangle in the well-known manner. The complete
n 2
180 APOLLONIUS OF PERGA
treatment of this problem in all its particular cases with their
Swpio-fxoi could present no difficulty to Apollonius.
If the two straight lines are parallel, the solution of the
problem gives a means of drawing any number of tangents
to an ellipse when two parallel tangents, their points of con-
tact, and the length of the parallel semi-diameter are given
(see Conies, III. 42). In the case of the hyperbola (III. 43)
the intercepts made by any tangent on the asymptotes contain
a constant rectangle. Accordingly the drawing of tangents
depends upon the particular case of our problem in which both
fixed points are the intersection of the two fixed lines.
(y) On determinate section (SLoopicr/ievr] rofirj), two Books.
The general problem here is, Given four points A, B, G, D on
a straight line, to determine another point P on the same
straight line such that the ratio AP . CP : BP . DP has a
given value. It is clear from Pappus's account x of the contents
of this work, and from his extensive collection of lemmas to
the different propositions in it, that the question was very
exhaustively discussed. To determine P by means of the
equation
AP.GP=\.BP.DP,
where A, B, C, D, A are given, is in itself an easy matter since
the problem can at once be put into the form of a quadratic
equation, and the Greeks would have no difficulty in reducing
it to the usual application of areas. If, however (as we may
fairly suppose), it was intended for application in further
investigations, the complete discussion of it would naturally
include not only the finding of a solution, but also the deter-
mination of the limits of possibility and the number of possible
solutions for different positions of the point-pairs A, C and
B, D, for the cases in which the points in either pair coincide,
or in which one of the points is infinitely distant, and so on.
This agrees with what we find in Pappus, who makes it clear
that, though we do not meet with any express mention of
series of point-pairs determined by the equation for different
values of A, yet the treatise contained what amounts to a com-
1 Pappus, vii, pp. 642-4.
ON DETERMINATE SECTION 181
plete Theory of Involution. Pappus says that the separate
cases were dealt with in which the given ratio was that of
either (1) the square of one abscissa measured from the
required point or (2) the rectangle contained by two such
abscissae to any one of the following:- (1) the square of one
abscissa, (2) the rectangle contained by one abscissa and
another separate line of given length independent of the
position of the required point, (3) the rectangle contained by
two abscissae. We learn also that maxima and minima were
investigated. From the lemmas, too, we may draw other
conclusions, e. g.
(1) that, in the case where A = 1, or AP .OP = BP .DP,
Apollonius used the relation BP .DP = AB . BO : AD . DO,
(2) that Apollonius probably obtained a double point E of the
involution determined by the point-pairs A, and B, D by
means of the relation
AB.BC.AD. DC = BE 2 : DE\
A possible application of the problem was the determination
of the points of intersection of the given straight line with a
conic determined as a four-line locus, since A, B, C, D are in
fact the points of intersection of the given straight line with
the four lines to which the locus has reference.
(S) On Contacts or Ta agencies {kirafyai), two Books.
Pappus again comprehends in one enunciation the varieties
of problems dealt with in the treatise, which we may repro-
duce as follows : Given three things, each of which may be
either a 'point, a straight line or a circle, to draw a circle
which shall pass through each of the given points (so far as it
is points that are given) and touch the straight lines or
circles. 1 The possibilities as regards the different data are
ten. We may have any one of the following: (1) three
points, (2) three straight lines, (3) two points and a straight
line, (4) two straight lines and a point, (5) two points and
a circle, (6) two circles and a point, (7) two straight lines and
1 Pappus, vii, p. 644, 25-8.
182 APOLLONIUS OF PERGA
a circle, (8) two circles and a straight line, (9) a point, a circle
and a straight line, (10) three circles. Of these varieties the
first two are treated in Eucl. IV ; Book I of Apollonius's
treatise treated of (3), (4), (5), (6), (8), (9), while (7), the case of
two straight lines and a circle, and (10), that of the three
circles, occupied the whole of Book II.
The last problem (10), where the data are three circles,
has exercised the ingenuity of many distinguished geometers,
including Vieta and Newton. Vieta (1540-1603) set the pro-
blem to Adrianus Komanus (van Roomen, 1561-1615) who
solved it by means of a hyperbola. Vieta was not satisfied
with this, and rejoined with his A r pollonius Gallus (1600) in
which he solved the problem by plane methods. A solution
of the same kind is given by Newton in his Arithmetica
Universalis (Prob. xlvii), while an equivalent problem is
solved by means of two hyperbolas in the Principia, Lemma
xvi. The problem is quite capable of a ' plane ' solution, and,
as a matter of fact, it is not difficult to restore the actual
solution of Apollonius (which of course used the 'plane' method
depending on the straight line and circle only), b}^ means of
the lemmas given by Pappus. Three things are necessary to
the solution. (1) A proposition, used by Pappus elsewhere 1
and easily proved, that, if two circles touch internally or
externally, any straight line through the point of contact
divides the circles into segments respectively similar. (2) The
proposition that, given three circles, their six centres of simili-
tude (external and internal) lie three by three on four straight
lines. This proposition, though not proved in Pappus, was
certainly known to the ancient geometers; it is even possible
that Pappus omitted to prove it because it was actually proved
by Apollonius in his treatise. (3) An auxiliary problem solved
by Pappus and enunciated by him as follows. 2 Given a circle
ABC, and given three points D, E, F in a straight line, to
inflect (the broken line) DAE (to the circle) so as to make BG
in a straight line with CF; in other words, to inscribe in the
circle a triangle the sides of which, when produced, pass
respectively through three given points lying in a straight
line. This problem is interesting as a typical example of the
ancient analysis followed by synthesis. Suppose the problem
1 Pappus, iv, pp. 194-6. 2 lb. vii, p.
ON CONTACTS OR TANGENCIES 183
solved, i.e. suppose DA, FA drawn to the circle cutting it in
points B, C such that BC produced passes through F.
Draw BG parallel to DF; join GC
and produce it to meet DF in H.
Then
IBAC= IBGC
= ICHF
= supplement of Z CHD ;
therefore A, D, H, C lie on a circle, and
DF.FH=AF.FC.
Now AE .EC is given, being equal to the square on the
tangent from E to the circle ; and DF is given ; therefore HE
is given, and therefore the point H.
But F is also given ; therefore the problem is reduced to
drawing HC, FC to meet the circle in such a way that, if
HC, FC produced meet the circle again in G, B, the straight
line BG is parallel to HF: a problem which Pappus has
previously solved. 1
Suppose this done, and draw BK the tangent at B meeting
HF in K. Then
Z KBC — ABGC, in the alternate segment,
= ICHF.
Also the angle CFK is common to the two triangles KBF,
CHF\ therefore the triangles are similar, and
CF:FH = KF:FB,
or HF.FK = BF.FC.
Now BF .FC is given, and so is HF;
therefore FK is given, and therefore K is given.
The synthesis is as follows. Take a point H on DE such
that DE . EH is equal to the square on the tangent from E to
the circle.
Next take K on HF such that HF . FK = the square on the
tangent from F to the circle.
Draw the tangent to the circle from K, and let B be the
point of contact. Join BF meeting the circle in C, and join
1 Pappus, vii, pp. 830-2.
184
APOLLONIUS OF PERGA
HG meeting the circle again in G. It is then easy to prove
that BG is parallel to DF.
Now join EG, and produce it to meet the circle again at A ;
join AB.
We have only to prove that AB, BD are in one straight line.
Since DE . EH = AE .EG, the points A, D, H, G are con-
cyclic.
Now the angle GHF, which is the supplement of the angle
GHD, is equal to the angle BGG, and therefore to the
angle BAG.
Therefore the angle BAG is equal to the supplement of
angle DUG, so that the angle BAG is equal to the angle DAG,
and AB, BD are in a straight line.
The problem of Apollonius is now easy. We will take the
case in which the required circle touches all the three given
circles externally as shown in the figure. Let the radii of the
ON CONTACTS OR TANGENCIES 185
given circles be a, b, c and their centres A, B, C. Let D, E, F
be the external centres of similitude so that BD : DC— b : c, &c.
Suppose the problem solved, and let P, Q, R be the points
of contact. Let PQ produced meet the circles with centres
A, B again in K, L. Then, by the proposition (1) above, the
segments KGP, QHL are both similar to the segment PYQ ;
therefore they are similar to one another. It follows that PQ
produced beyond L passes through F. Similarly QR, PR
produced pass respectively through D, E.
Let PE, QD meet the circle with centre C again in M, N.
Then, the segments PQR, RNM being similar, the angles
PQR, RNM are equal, and therefore MN is parallel to PQ.
Produce NM to meet EF in V.
Then EV:EF = EM: EP = EC:EA = c:a;
therefore the point V is given.
Accordingly the problem reduces itself to this : Given three
points V, E, D in a straight line, it is required to draw DR, ER
to a point R on the circle with centre C so that, if DR, ER meet
the circle again in N, M, NM produced shall pass through V.
This is the problem of Pappus just solved.
Thus R is found, and DR, ER produced meet the circles
with centres B and A in the other required points Q, P
respectively.
(e) Plane loci, two Books.
Pappus gives a pretty full account of the contents of this
work, which has sufficed to enable restorations of it to
be made by three distinguished geometers, Fermat, van
Schooten, and (most completely) by Robert Simson. Pappus
prefaces his account by a classification of loci on two
different plans. Under the first classification loci are of three
kinds: (1) efeKTiKoi, holding-in or fixed; in this case the
locus of a point is a point, of a line a line, and of a solid
a solid, where presumably the line or solid can only move on
itself so that it does not change its position: (2) Siego-
Slkol, pasdng-along : this is the ordinary sense of a locus,
where the locus of a point is a line, and of a line a solid:
(3) dvao-Tpo(f)iKoi, moving backvjards and forwards, as it were,
in which sense a plane may be the locus of a point and a solid
186 AP0LL0N1US OF PERGA
of a line. 1 The second classification is the familiar division into
'plane, solid, and linear loci, plane loci being straight lines
and circles only, solid loci conic sections only, and linear loci
those which are not straight lines nor circles nor any of the
conic sections. The loci dealt with in our treatise are accord-
ingly all straight lines or circles. The proof of the pro-
positions is of course enormously facilitated by the use of
Cartesian coordinates, and many of the loci are really the
geometrical equivalent of fundamental theorems in analytical
or algebraical geometry. Pappus begins with a composite
enunciation, including a number of propositions, in these
terms, which, though apparently confused, are not difficult
to follow out:
4ft
1 If two straight lines be drawn, from one given point or from
two, which are (a) in a straight line or (b) parallel or
(c) include a given angle, and either (a) bear a given ratio to
one another or (/?) contain a given rectangle, then, if the locus
of the extremity of one of the lines is a plane locus given in
position, the locus of the extremity of the other will also be a
plane locus given in position, which will sometimes be of the
same kind as the former, sometimes of the other kind, and
will sometimes be similarly situated with reference to the
straight line, and sometimes contrarily, according to the
particular differences in the suppositions.' 2
(The words ' with reference to the straight line ' are obscure, but
the straight line is presumably some obvious straight line in
each figure, e. g., when there are two given points, the straight
line joining them.) After quoting three obvious loci ' added
by Charmandrus ', Pappus gives three loci which, though con-
taining an unnecessary restriction in the third case, amount
to the statement that any equation of the first degree between
coordinates inclined at fixed angles to (a) two axes perpen-
dicular or oblique, (h) to any number of axes, represents a
straight line. The enunciations (5-7) are as follows. 3
5. ' If, when a straight line is given in magnitude and is
moved so as always to be parallel to a certain straight line
given in position, one of the extremities (of the moving
straight line), lies on a straight line given in position, the
1 Pappus, vii, pp. 660. 18-662. 5. 2 16,'vii, pp. 662. 25-664. 7.
3 lb., pp. 664. 20-666. 6.
PLANE LOCI 187
other extremity will also lie on a straight line given in
position.'
(That is, x = a or y = b in Cartesian coordinates represents a
straight line.)
6. ' If from any point straight lines be drawn to meet at given
angles two straight lines either parallel or intersecting, and if
the straight lines so drawn have a given ratio to one another
or if the sum of one of them and a line to which the other has
a given ratio be given (in length), then the point will lie on a
straight line given in position.'
(This includes the equivalent of saying that, if x, y be the
coordinates of the point, each of the equations x = my,
x + my = c represents a straight line.)
7. ' If any number of straight lines be given in position, and
straight lines be drawn from a point to meet them at given
angles, and if the straight lines so drawn be such that the
rectangle contained by one of them and a given straight line
added to the rectangle contained by another of them and
(another) given straight line is equal to the rectangle con-
tained by a third and a (third) given straight line, and simi-
larly with the others, the point will lie on a straight line given
in position.'
(Here we have trilinear or multilinear coordinates propor-
tional to the distances of the variable point from each of the
three or more fixed lines. When there are three fixed lines,
the statement is that ax + by = cz represents a straight line.
The precise meaning of the words 'and similarly with the
the others ' or ' of the others ' — kolI tw \olttS)v 6/ioico? — -is
uncertain ; the words seem to imply that, when there were
more than three rectangles ax, by, cz . . . , two of them were
taken to be equal to the sum of all the others ; but it is quite
possible that Pappus meant that any linear equation between
these rectangles represented a straight line. Precisely how
far Apollonius went in generality we are not in a position to
judge.)
The last enunciation (8) of Pappus referring to Book I
states that,
' If from any point (two) straight lines be drawn to meet (two)
parallel straight lines given in position at given angles, and
188 APOLLONIUS OF PERGA
cut off from the parallels straight lines measured from given
points on them such that (a) they have a given ratio or
(b) they contain a given rectangle or (c) the sum or difference
of figures of given species described on them respectively is
equal to a given area, the point will lie on a straight line
given in position.' 1
The contents of Book II are equally interesting. Some of
the enunciations shall for brevity be given by means of letters
instead of in general terms. If from two given points A, B
two straight lines be ' inflected ' (KXacrOaxriu) to a point P, then
(1), if AP 2 ^ BP 2 is given, the locus of P is a straight line;
(2) if AP, BP are in a given ratio, the locus is a straight line
or a circle [this is the proposition quoted by Eutocius in his
commentary on the Conies, but already known to Aristotle] ;
(4) if AP 2 is ' greater b}^ a given area than in a given ratio '
to BP 2 , i.e. if AP 2 = a 2 + m . BP 2 , the locus is a circle given in
position. An interesting proposition is (5) that, ' If from any
number of given points whatever straight lines be inflected to
one point, and the figures (given in species) described on all of
them be together equal to a given area, the point will lie on
a circumference (circle) given in position ' ; that is to say, if
a . AP 2 + fi . BP 2 + y . CP 2 + ... = a given area (where a, ft, y . . .
are constants), the locus of P is a circle. (3) states that, if
AN be a fixed straight line and A a fixed point on it, and if
AP be any straight line drawn to a point P such that, if PN
is perpendicular to AN, AP 2 — a . AN or a . BN, where a- is a
given length and B is another fixed point on AN, then the
locus of P is a circle given in position ; this is equivalent
to the fact that, if A be the origin, AN the axis of x, and
x = AN,y = PN be the coordinates of P, the locus x 2 + y 2 = ax
or x 2 + y 2 = a(x — b) is a circle. (6) is somewhat obscurely
enunciated : ' If from two given points straight lines be in-
flected (to a point), and from the point (of concourse) a straight
line be drawn parallel to a straight line given in position and
cutting off from another straight line given in position an
intercept measured from a given point on it, and if the sum of
figures (given in species) described on the two inflected lines
be equal to the rectangle contained by a given straight line
and the intercept, the point at which the straight lines are
1 Pappus, vii, p. 666. 7-13.
PLANE LOCI 189
inflected lies on a circle given in position.' The meaning-
seems to be this : Given two fixed points A, B y a length a,
a straight line OX with a point fixed upon it, and a direc-
tion represented, say, by any straight line OZ through 0, then,
if AP, BP be drawn to P, and PM parallel to OZ meets OX
in M, the locus of P will be a circle given in position if
a.AP 2 + /3.BP 2 = a.0M,
where a, /3 are constants. The last two loci are again
obscurely expressed, but the sense is this : (7) If PQ be any
chord of a circle passing through a fixed internal point 0, and
R be an external point on PQ produced such that either
(a) OR 2 = PR.RQ or (b) 0R 2 + P0 . 0Q= PR . RQ, the locus
of R is a straight line given in position. (8) is the reciprocal
of this : Given the fixed point 0, the straight line which is
the locus of R, and also the relation (a) or (b), the locus of
P, Q is a circle.
(£) Nevcreis (Vergings or Inclinations), two Books.
As we have seen, the problem in a vevo-is is to place
between two straight lines, a straight line and a curve, or
two curves, a straight line of given length in such a way
that it verges towards a fixed point, i.e. it will, if pro-
duced, pass through a fixed point. Pappus observes that,
when we come to particular cases, the problem will be
' plane ', ' solid ' or ' linear ', according to the nature of the
particular hypotheses ; but a selection had been made from
the class which could be solved by plane methods, i.e. by
means of the straight line and circle, the object being to give
those which were more generally useful in geometry. The
following were the cases thus selected and proved. 1
I. Given (a) a semicircle and a straight line at right angles
to the base, or (b) two semicircles with their bases in a straight
line, to insert a straight line of given length verging to an
angle of the semicircle [or of one of the semicircles].
II. Given a rhombus with one side produced, to insert
a straight line of given length in the external angle so that it
verges to the opposite angle.
1 Pappus, vii, pp. 670-2.
190 APOLLONIUS OF PERGA
III. Given a circle, to insert a chord of given length verging
to a given point.
In Book I of Apollonius's work there were four cases of
I (a), two cases of III, and two of II ; the second Book con-
tained ten cases of I (b).
Restorations were attempted by Marino Ghetaldi (Apollonius
redivivus, Venice, 1607, and Apollonius redivivus . . . Liber
secundus, Venice, 1613), Alexander Anderson (in a Supple-
onentum Apollonii redivivi, 1612), and Samuel Horsley
(Oxford, 1770); the last is much the most complete.
In the case of the rhombus (II) the construction of Apollonius
can be restored with certainty. It depends on a lemma given
by Pappus, which is as follows : Given a rhombus AD with
diagonal BG produced to E, if F be taken on BG such that EF
is a mean proportional between BE and EG, and if a circle be
described with E as centre and EF as radius cutting CD
in K and AG produced in H, then shall B, K, H be in one
straight line. 1
Let the circle cut AG in L, join LK meeting BG in M, and
join HE, LE, KE.
Since now CL, GK are equally inclined to the diameter of
the circle, CL = GK. Also EL = EK, and it follows that the
triangles ECK, ECL are equal in all respects, so that
, Z CKE = L CLE = Z CHE.
By hypothesis, EB:EF=EF: EC,
or EB:EK = EK:EC.
1 Pappus, vii, pp. 778-80.
NET2EI2 (VERGINGS OR INCLINATIONS) 191
Therefore the triangles BEK, KEG, which have the angle
BEK common, are similar, and
Z GBK = Z GKE = Z GEE (from above).
But Z HGE = IAGB= Z BCK.
Therefore in the triangles CBK, GHE two angles are
respectively equal, so that Z GEH — Z GKB also.
But since LGKE = I CHE (from above), K, C, E, E are
concyclic.
Hence Z CEH+ Z GKE = (two right angles) ;
therefore, since Z GEE — Z GKB,
Z GKB + Z Cif # = (two right angles),
and BKE is a straight line.
It is certain, from the nature of this lemma, that Apollonius
made his construction by drawing the circle shown in the
figure.
He would no doubt arrive at it by analysis somewhat as
follows.
Suppose the problem solved, and EK inserted as re-
quired (= h).
Bisect EK in N, and draw NE at right angles to KE
meeting BC produced in E. Draw KM perpendicular to BC,
and produce it to meet AC in L. Then, by the property of
the rhombus, LM = MK, and, since KN = NE also, MN is
parallel to LE.
Now, since the angles at M, N are right, M, K, N, E are
concyclic.
Therefore ICEK = Z.MNK = IGEK, so that C, K, E, E
are concyclic.
Therefore Z BCD = supplement of KCE = LEEK = lEKE,
and the triangles EKE, DGB are similar.
Lastly,
IEBK=IEKE-ICEK=IEEK-ICEK=IEEC=IEKC;
therefore the triangles EBK, EKC are similar, and
BE:EK = EK:EC,
or BE.EC = EK 2 .
192 APOLLONIUS OF PERGA
But, by similar triangles EKH, DCB,
EK:KH=DC:CB,
and, since the ratio DC:CB, as well as KH, is given, EK
is given.
The construction then is as follows.
If k be the given length, take a straight line p such that
p:k = AB:BC:
apply to BG a rectangle BE . EC equal to p 1 and exceeding by
a square ; then with E as centre and radius equal to p describe a
circle cutting AC produced in H and CD in K. HK is then
equal to k and, by Pappus's lemma, verges towards B.
Pappus adds an interesting solution of the same problem
with reference to a square instead of a rhombus ; the solution
is by one Heraclitus and depends on a lemma which Pappus
also gives. 1
We hear of yet other lost works by Apollonius.
(rj) A Comparison of the dodecahedron with the icosahedron.
This is mentioned by Hypsicles in the preface to the so-called
Book XIV of Euclid. Like the Conies, it appeared in two
editions, the second of which contained the proposition that,
if there be a dodecahedron and an icosahedron inscribed in
one and the same sphere, the surfaces of . the solids are in the
same ratio as their volumes ; this was established by showing
that the perpendiculars from the centre of the sphere to
a pentagonal face of the dodecahedron and to a triangular
face of the icosahedron are equal.
(0) Marinus on Euclid's Data speaks of a General Treatise
(kcc66\ov Trpay/jLCLTeia) in which Apollonius used the word
assigned (TtTayfiivov) as a comprehensive term to describe the
datum in general. It would appear that this work must
have dealt with the fundamental principles of mathematics,
definitions, axioms, &c, and that to it must be referred the
various remarks on such subjects attributed to Apollonius by
Proclus, the elucidation of the notion of a line, the definition
1 Pappus, vii, pp. 780-4.
OTHER LOST WORKS 193
of plane and solid angles, and his attempts to prove the axioms ;
it must also have included the three definitions (13-15) in
Euclid's Data which, according to a scholium, were due to
Apollonius and must therefore have been interpolated (they
are definitions of KarrjyfjLevr), dvrjy/ievr], and the elliptical
phrase irapa Oeaei, which means 'parallel to a straight line
given in position '). Probably the same work also contained
Apollonius's alternative constructions for the problems of
Eucl. I. 10, 11 and 23 given by Proclus. Pappus speaks
of a mention by Apollonius ' before his own elements ' of the
class of locus called e0e/cri/coy, and it may be that the treatise
now in question is referred to rather than the Plane Loci
itself.
(i) The work On the Cochlias was on the cylindrical helix.
It included the theoretical generation of the curve on the
surface of the cylinder, and the proof that the curve is
homoeomeric or uniform, i.e. such that any part will fit upon
or coincide with any other.
(k) A work on Unordered Irrationals is mentioned by
Proclus, and a scholium on Eucl. X. 1 extracted from Pappus's
commentary remarks that ' Euclid did not deal with all
rationals and irrationals, but only with the simplest kinds by
the combination of which an infinite number of irrationals
are formed, of which latter Apollonius also gave some '.
To a like effect is a passage of the fragment of Pappus's
commentary on Eucl. X discovered in an Arabic translation
by Woepcke : c it was Apollonius who, besides the ordered
irrational magnitudes, showed the existence of the unordered,
and by accurate methods set forth a great number of them '.
The hints given by the author of the commentary seem to imply
that Apollonius's extensions of the theory of irrationals took
two directions, (1) generalizing the medial straight line of
Euclid, on the basis that, between two lines commensurable in
square (only), we may take not only one sole medial line but
three or four, and so on ad infinitum, since we can take,
between any two given straight lines, as many lines as
we please in continued proportion, (2) forming compound
irrationals by the addition and subtraction of more than two
terms of the sort composing the binomials, apotomes, &c.
1523.2 O
194 APOLLONIUS OF PERGA
(A) On the burning-mirror {rrepl rod Trvptov) is the title of
another work of Apollonius mentioned by the author of the
Fragmentum mathematicum Bobiense, which is attributed by
Heiberg to Anthemius but is more likely (judging by its sur-
vivals of antiquated terminology) to belong to a much earlier
date. The fragment shows that Apollonius discussed the
spherical form of mirror among others. Moreover, the extant
fragment by Anthemius himself (on burning mirrors) proves the
property of mirrors of parabolic section, using the properties of
the parabola (a) that the tangent at any point makes equal
angles with the axis and with the focal distance of the point,
and (b) that the distance of any point on the curve from the
focus is equal to its distance from a certain straight line
(our ' directrix ') ; and we can well believe that the parabolic
form of mirror was also considered in Apollonius's work, and
that he was fully aware of the focal properties of the parabola,
notwithstanding the omission from the Conies of all mention
of the focus of a parabola.
(//) In a work called (Lkvtoklov (' quick-delivery ') Apollonius
is said to have found an approximation to the value of rr ' by
a different calculation (from that of Archimedes), bringing it
within closer limits '} Whatever these closer limits may have
been, they were considered to be less suitable for practical use
than those of Archimedes.
It is a moot question whether Apollonius's system of arith-
metical notation (by tetrads) for expressing large numbers
and performing the usual arithmetical operations with them,
as described by Pappus, was included in this same work.
Heiberg thinks it probable, but there does not seem to be any
necessary reason why the notation for large numbers, classify-
ing them into myriads, double myriads, triple myriads, &c,
i.e. according to powers of 10,000, need have been connected
with the calculation of the value of ir, unless indeed the num-
bers used in the calculation were so large as to require the
tetradic system for the handling of them.
We have seen that Apollonius is credited with a solu-
tion of the problem of the two mean proportionals (vol. i,
pp. 262-3).
1 v. Eutocius on Archimedes, Measurement of a Circle,
OTHER LOST WORKS 195
Astronomy.
We are told by Ptolemaeus Chennus * that Apollonius was
famed for his astronomy, and was called e (Epsilon) because
the form of that letter is associated with that of the moon, to
which his accurate researches principally related. Hippolytus
says he made the distance of the moon's circle from the sur-
face of the earth to be 500 myriads of stades. 2 This figure
can hardly be right, for, the diameter of the earth being,
according to Eratosthenes's evaluation, about eight myriads of
stades, this would make the distance of the moon from the
earth about 125 times the earth's radius. This is an unlikely
figure, seeing that Aristarchus had given limits for the ratios
between the distance of the moon and its diameter, and
between the diameters of the moon and the earth, which lead
to about 1 9 as the ratio of the moon's distance to the earth's
radius. Tannery suggests that perhaps Hippolytus made a
mistake in copying from his source and took the figure of
5,000,000 stades to be the length of the radius instead of the
diameter of the moon's orbit.
But we have better evidence of the achievements of Apol-
lonius in astronomy. In Ptolemy's Syntaxis 3 he appears as
an authority upon the hypotheses of epicycles and eccentrics
designed to account for the apparent motions of the planets.
The propositions of Apollonius quoted by Ptolemy contain
exact statements of the alternative hypotheses, and from this
fact it was at one time concluded that Apollonius invented
the two hypotheses. This, however, is not the case. The
hypothesis of epicycles was already involved, though with
restricted application, in the theory of Heraclides of Pontus
that the two inferior planets, Mercury and Venus, revolve in
circles like satellites round the sun, while the sun itself
revolves in a circle round the earth ; that is, the two planets
describe epicycles about the material sun as moving centre.
In order to explain the motions of the superior planets by
means of epicycles it was necessary to conceive of an epicycle
about a point as moving centre which is not a material but
a mathematical point. It was some time before this extension
of the theory of epicycles took place, and in the meantime
1 apud Photium, Cod. cxc, p. 151 b 18, ed. Bekker.
2 Hippol. Refut. iv. 8, p. 66, ed. Duncker. 3 Ptolemy, Syntaxis, xii. 1.
o 2
196 APOLLONIUS OF PERGA
another hypothesis, that of eccentrics, was invented to account
for the movements of the superior planets only. We are at this
stage when we come to Apollonius. His enunciations show
that he understood the theory of epicycles in all its generality,
but he states specifically that the theory of eccentrics can only
be applied to the three planets which can be at any distance
from the sun. The reason why he says that the eccentric
hypothesis will not serve for the inferior planets is that, in
order to make it serve, we should have to suppose the circle
described by the centre of the eccentric circle to be greater
than the eccentric circle itself. (Even this generalization was
made later, at or before the time of Hipparchus.) Apollonius
further says in his enunciation about the eccentric that ' the
centre of the eccentric circle moves about the centre of the
zodiac in the direct order of the signs and at a s r peed equal to
that of the sun, while the star moves on the eccentric about
its centre in the inverse order of the signs and at a speed
equal to the anomaly \ It is clear from this that the theory
of eccentrics was invented for the specific purpose of explain-
ing the movements of Mars, Jupiter, and Saturn about the
sun and for that purpose alone. This explanation, combined
with the use of epicycles about the sun as centre to account
for the motions of A r enus and Mercury, amounted to the
system of Tycho Brahe ; that system was therefore anticipated
by some one intermediate in date between Heraclides and
Apollonius and probably nearer to the latter, or it may
have been Apollonius himself who took this important step.
If it was, then Apollonius, coming after Aristarchus of
Samos, would be exactly the Tycho Brahe of the Copernicus
of antiquity. The actual propositions quoted by Ptolemy as
proved by Apollonius among others show mathematically at
what points, under each of the two hypotheses, the apparent
forward motion changes into apparent retrogradation and
vice versa, or the planet appears to be stationary.
XV
THE SUCCESSORS OF THE GREAT GEOMETERS
With Archimedes and Apollonius Greek geometry reached
its culminating point. There remained details to be filled
in, and no doubt in a work such as, for instance, the Conies
geometers of the requisite calibre could have found proposi-
tions containing the germ of theories which were capable of
independent development. But, speaking generally, the fur-
ther progress of geometry on general lines was practically
barred by the restrictions of method and form which were
inseparable from the classical Greek geometry. True, it was
open to geometers to discover and investigate curves of a
higher order than conies, such as spirals, conchoids, and the
like. But the Greeks could not get very far even on these
lines in the absence of some system of coordinates and without
freer means of manipulation such as are afforded by modern
algebra, in contrast to the geometrical algebra, which could
only deal with equations connecting lines, areas, and volumes,
but involving no higher dimensions than three, except in so
far as the use of proportions allowed a very partial exemp-
tion from this limitation. The theoretical methods available
enabled quadratic, cubic and bi-quadratic equations or their
equivalents to be solved. But all the solutions were geometri-
cal ; in other words, quantities could only be represented by
lines, areas and volumes, or ratios between them. There was
nothing corresponding to operations with general algebraical
quantities irrespective of what they represented. There were
no symbols for such quantities. In particular, the irrational
was discovered in the form of incommensurable lines ; hence
irrationals came to be represented by straight lines as they
are in Euclid, Book X, and the Greeks had no other way of
representing them. It followed that a product of two irra-
tionals could only be represented by a rectangle, and so on.
Even when Diophantus came to use a symbol for an unknown
198 SUCCESSORS OF THE GREAT GEOMETERS
quantity, it was only an abbreviation for the word dpiOfios,
with the meaning of ' an undetermined multitude of units ',
not a general quantity. The restriction then of the algebra
employed by geometers to the geometrical form of algebra
operated as an insuperable obstacle to any really new depar-
ture in theoretical geometry.
It might be thought that tbere was room for further exten-
sions in the region of solid geometry. But the fundamental
principles of solid geometry had also been laid down in Euclid,
Books XI-XIII ; the theoretical geometry of the sphere had
been fully treated in the ancient spkaeric ; and any further
application of solid geometry, or of loci in three dimensions,
was hampered by the same restrictions of method which
hindered the further progress of plane geometry.
Theoretical geometry being thus practically at the end of
its resources, it was natural that mathematicians, seeking for
an opening, should turn to the applications of geometry. One
obvious branch remaining to be worked out was the geometry
of measurement, or mensuration in its widest sense, which of
course had to wait on pure theory and to be based on its
results. One species of mensuration was immediately required
for astronomy, namely the measurement of triangles, especially
spherical triangles ; in other words, trigonometry plane and
spherical. Another species of mensuration was that in which
an example had already been set by Archimedes, namely the
measurement of areas and volumes of different shapes, and
arithmetical approximations to their true values in cases
where they involved surds or the ratio (it) between the
circumference of a circle and its diameter ; the object of such
mensuration was largely practical. Of these two kinds of
mensuration, the first (trigonometry) is represented by Hip-
parchus, Menelaus and Ptolemy ; the second by Heron of
Alexandria. These mathematicians, will be dealt with in later
chapters ; this chapter will be devoted to the successors of the
great geometers who worked on the same lines as the latter.
Unfortunately we have only very meagre information as to
what these geometers actually accomplished in keeping up the
tradition. No geometrical works by them have come down
to us in their entirety, and we are dependent on isolated
extracts or scraps of information furnished by commen-
NICOMEDES 199
tators, and especially by Pappus and Eutocius. Some of
these are very interesting, and it is evident from the
extracts from the works of such writers as Diodes and
Dionysodorus that, for some time after Archimedes and
Apollonius, mathematicians had a thorough grasp of the
contents of the works of the great geometers, and were able
to use the principles and methods laid down therein with
ease and skill.
Two geometers properly belonging to this chapter have
already been dealt with. The first is Nicomedes, the inventor
of the conchoid, who was about intermediate in date between
Eratosthenes and Apollonius. The conchoid has already been
described above (vol. i, pp. 238-40). It gave a general method
of solving any vevcris where one of the lines which cut off an
intercept of given length on the line verging to a given point
is a straight line ; and it was used both for the finding of two
mean proportionals and for the trisection of any angle, these
problems being alike reducible to a vevo-is of this kind. How
far Nicomedes discussed the properties of the curve in itself
is uncertain ; we only know from Pappus that he proved two
properties, (1) that the so-called 'ruler' in the instrument for
constructing the curve is an asymptote, (2) that any straight
line drawn in the space between the ' ruler ' or asymptote and
the conchoid must, if produced, be cut by the conchoid. 1 The
equation of the curve referred to polar coordinates is, as we
have seen, r = a + b sec 6. According to Eutocius, Nicomedes
prided himself inordinately on his discovery of this curve,
contrasting it with Eratosthenes's mechanism for finding any
number of mean proportionals, to which he objected formally
and at length on the ground that it was impracticable and
entirely outside the spirit of geometry. 2
Nicomedes is associated by Pappus with Dinostratus, the
brother of Menaechmus, and others as having applied to the
squaring of the circle the curve invented by Hippias and
known as the quadratrix, z which was originally intended for
the purpose of trisecting any angle. These facts are all that
we know of Nicomedes's achievements.
1 Pappus, iv, p. 244. 21-8.
2 Eutoc. on Archimedes, On the Sphere and Cylinder, Archimedes,
vol. iii, p. 98.
3 Pappus, iv, pp. 250. 33-252. 4. Cf. vol. i, p. 225 sq.
200 SUCCESSORS OF THE GREAT GEOMETERS
The second name is that of Diocles. We have already
(vol. i, pp. 264-6) seen him as the discoverer of the curve
known as the cissoid, which he used to solve the problem
of the two mean proportionals, and also (pp. 47-9 above)
as the author of a method of solving the equivalent of
a certain cubic equation by means of the intersection
of an ellipse and a hyperbola. We are indebted for our
information on both these subjects to Eutocius, 1 who tells
us that the fragments which he quotes came from Diocles's
work rrepl nuptiais, On burning -mirrors. The connexion of
the two things with the subject of this treatise is not obvious,
and we may perhaps infer that it was a work of considerable
scope. What exactly were the forms of the burning-mirrors
discussed in the treatise it is not possible to say, but it is
probably safe to assume that among them were concave
mirrors in the forms (1) of a sphere, (2) of a paraboloid, and
(3) of the surface described by the revolution of an ellipse
about its major axis. The author of the Fragmentum mathe-
maticum Bobiense says that Apollonius in his book On the
burning -mirror discussed the case of the concave spherical
mirror, showing about what point ignition would take place ;
and it is certain that Apollonius was aware that an ellipse has
the property of reflecting all rays through one focus to the
other focus. Nor is it likely that the corresponding property
of a parabola with reference to rays parallel to the axis was
unknown to Apollonius. Diocles therefore, writing a century
or more later than Apollonius, could hardly have failed to
deal with all three cases. True, Anthemius (died about
A. D. 534) in his fragment on burning-mirrors says that the
ancients, while mentioning the usual burning-mirrors and
saying that such figures are conic sections, omitted to specify
which conic sections, and how produced, and gave no geo-
metrical proofs of their properties. But if the properties
were commonly known and quoted, it is obvious that they
must have been proved by the ancients, and the explanation
of Anthemius's remark is presumably that the original works
in which they were proved (e.g. those of Apollonius and
Diocles) were already lost when he wrote. There appears to
be no trace of Diocles's work left either in Greek or Arabic,
1 Eutocius, loc. cit., p. 66. 8 sq., p. 160. 3 sq.
DIOCLES 201
unless we have a fragment from it in the Fragmentum
mathematicum Bobieme. But Moslem writers regarded Diocles
as the discoverer of the parabolic burning-mirror; 'the ancients',
says al Singari (Sachawi, Ansarl), ' made mirrors of plane
surfaces. Some made them concave (i.e. spherical) until
Diocles (Diiiklis) showed and proved that, if the surface of
these mirrors has its curvature in the form of a parabola, they
then have the greatest power and burn most strongly. There
is a work on this subject composed by Ibn al-Haitham.' This
work survives in Arabic and in Latin translations, and is
reproduced by Heiberg and Wiedemann 1 ; it does not, how-
ever, mention the name of Diocles, but only those of Archi-
medes and Anthemius. Ibn al-Haitham says that famous
men like Archimedes and Anthemius had used mirrors made
up of a number of spherical rings ; afterwards, he adds, they
considered the form of curves which would reflect rays to one
point, and found that the concave surface of a paraboloid of
revolution has this property. It is curious to find Ibn al-
Haitham saying that the ancients had not set out the proofs
sufficiently, nor the method by which they discovered them,
words which almost exactly recall those of Anthemius himself.
Nevertheless the whole course of Ibn al-Haitham' s proofs is
on the Greek model, Apollonius being actually quoted by name
in the proof of the main property of the parabola required,
namely that the tangent at any point of the curve makes
equal angles with the focal distance of the point and the
straight line drawn through it parallel to the axis. A proof
of the property actually survives in the Greek Fragmentum
mathematicum Bobiense, which evidently came from some
treatise on the parabolic burning-mirror ; but Ibn al-Haitham
does not seem to have had even this fragment at his disposal,
since his proof takes a different course, distinguishing three
different cases, reducing the property by analysis to the
known property AN = AT, and then working out the syn-
thesis. The proof in the Fragmentum is worth giving. It is
substantially as follows, beginning with the preliminary lemma
that, if FT } the tangent at any point P, meets the axis at T x
and if AS be measured along the axis from the vertex A
equal to \AL y where AL is the parameter, then SF = ST
1 Bibliotheca mathematica, x 3 , 1910, pp. 201-37*
202 SUCCESSORS OF -THE GREAT GEOMETERS
Let PN be the ordinate from P ; draw A Y at right angles
to the axis meeting PT in Y, and join SY.
Now PN*=AL.AN
= 4 AS. AN
= 4AS.AT (since A# = AT).
But PlY = 2 AY (since JJV= JT) ;
therefore A Y 2 = TA . AS,
and the angle TYS is right.
The triangles SYT, SYP being right-angled, and TY being
equal to YP, it follows that SP = ST.
With the same figure, let BP be a ray parallel to AN
impinging on the curve at P. It is required to prove that
the angles of incidence and reflection (to S) are equal.
We have SP = ST, so that ' the angles at the points T, P
are equal. So ', says the author, ' are the angles TPA, KPR
[the angles between the tangent and the curve on each side of
the point of contact]. Let the difference between the angles
be taken ; therefore the angles SPA, RPB which remain
[again ' mixed ' angles] are equal. Similarly we shall show
that all the lines drawn parallel to .4$ will be reflected at
equal angles to the point S.'
The author then proceeds : ' Thus burning-mirrors con-
structed with the surface of impact (in the form) of the
section of a right-angled cone may easily, in the manner
DIOCLES. PERSEUS 203
above shown, be proved to bring about ignition at the point
indicated.'
Heiberg held that the style of this fragment is Byzantine
and that it is probably by Anthemius. Cantor conjectured
that here we might, after all, have an extract from Diocles's
work. Heiberg's supposition seems to me untenable because
of the author's use (1) of the ancient terms ' section of
a right-angled cone ' for parabola and ' diameter ' for axis
(to say nothing of the use of the parameter, of which there is
no word in the genuine fragment of Anthemius), and (2) of
the mixed ' angles of contact '. Nor does it seem likely that
even Diocles, living a century after Apollonius, would have
spoken of the 'section of a right-angled cone' instead of a
parabola, or used the ' mixed ' angle of which there is only the
merest survival in Euclid. The assumption of the equality
of the two angles made by the curve with the tangent on
both sides of the point of contact reminds us of Aristotle's
assumption of the equality of the angles ' of a segment ' of
a circle as prior to the truth proved in Eucl. I. 5. I am
inclined, therefore, to date the fragment much earlier even
than Diocles. Zeuthen suggested that the property of the
paraboloidal mirror may have been discovered by Archimedes,
who, according to a Greek tradition, wrote Gatoptrica. This,
however, does not receive any confirmation in Ibn al-Haitham
or in Anthemius, and we can only say that the fragment at
least goes back to an original which was probably not later
than Apollonius.
Perseus is only known, from allusions to him in Proclus, 1
as the discoverer and investigator of the spiric sections. They
are classed by Proclus among curves obtained by cutting
solids, and in this respect they are associated with the conic
sections. We may safely infer that they were discovered
after the conic sections, and only after the theory of conies
had been considerably developed. This was already the case
in Euclid's time, and it is probable, therefore, that Perseus was
not earlier than Euclid. On the other hand, by that time
the investigation of conies had brought the exponents of the
subject such fame that it would be natural for mathematicians
to see whether there was not an opportunity for winning a
1 Proclus on Eucl. I, pp. 111. 23-112. 8, 356. 12. Cf. vol. i, p. 226,
204 SUCCESSORS OF THE GREAT GEOMETERS
like renown as discoverers of other curves to be obtained by
cutting well-known solid figures other than the cone and
cylinder. A particular case of one such solid figure, the
cnreipa, had already been employed by Archytas, and the more
general form of it would not unnaturally be thought of as
likely to give sections worthy of investigation. Since Geminus
is Proclus's authority, Perseus may have lived at any date from
Euclid's time to (say) 75 B.C., but the most probable supposi-
tion seems to be that he came before Apollonius and .near to
Euclid in date.
The spire in one of its forms is what we call a tore, or an
anchor- ring. It is generated by the revolution of a circle
about a straight line in its plane in such a way that the plane
of the circle always passes through the axis of revolution. It
takes three forms according as the axis of revolution is
(a) altogether outside the circle, when the spire is open
(Sizyjis), (b) a tangent to the circle, when the surface is con-
tinuous (avvexvs)' or ( c ) a chord of the circle, when it is inter-
laced (efj.7r€7rXeyfiei/r]), or crossing -it self (kiraWdrTova-a) ; an
alternative name for the surface was KpiKos, a ring. i Perseus
celebrated his discovery in an epigram to the effect that
' Perseus on his discovery of three lines (curves) upon five
sections gave thanks to the gods therefor'. 1 There is some
doubt about the meaning of ' three lines upon five sections'
(Tpet? ypafipLas kirl irevre rouaTs). We gather from Proclus's
account of three sections distinguished by Perseus that the
plane of section was always parallel to the axis of revolution
or perpendicular to the plane which cuts the tore symmetri-
cally like the division in a split-ring. It is difficult to inter-
pret the phrase if it means three curves made by five different
sections. Proclus indeed implies that the three curves were
sections of the three kinds of tore respectively (the open, the
closed, and the interlaced), but this is evidently a slip.
Tannery interprets the phrase as meaning ' three curves in
addition to five sections '. 2 Of these the five sections belong
to the open tore, in which the distance (c) of the centre of the
generating circle from the axis of revolution is greater than
the radius (a) of the generating circle. If d be the perpen-
1 Proclus on Eucl. I, p. 112. 2.
2 See Tannery, Memoires scientifiques, II, pp. 24-8.
PERSEUS 205
dicular distance of the plane of section from the axis of rota-
tion, we can distinguish the following cases :
(1) c + a>d>c. Here the curve is an oval.
(2) d = c: transition from case (1) to the next case.
(3) od>c — a. The curve is now a closed curve narrowest
in the middle.
(4) d = c — a. In this case the curve is the hi r ppopede
(horse-fetter), a curve in the shape of the figure of 8. The
lemniscate of Bernoulli is a particular case of this curve, that
namely in which c = 2 a.
(5) c — a>d>0. In this case the section consists of two
ovals symmetrical with one another.
The three curves specified by Proclus are those correspond-
ing to (1), (3) and (4).
When the tore is ' continuous ' or closed, c = a, and we have
sections corresponding to (1), (2) and (3) only; (4) reduces to
two circles touching one another.
But Tannery finds in the third, the interlaced, form of tore
three new sections corresponding to (1) (2) (3), each with an
oval in the middle. This would make three curves in addi-
tion to the five sections, or eight curves in all. We cannot be
certain that this is the true explanation of the phrase in the
epigram ; but it seems to' be the best suggestion that has
been made.
According to Proclus, Perseus worked out the property of
his curves, as Nicomedes did that of the conchoid, Hippias
that of the quadratrix, and Apollonius those of the three
conic sections. That is, Perseus must have given, in some
form, the equivalent of the Cartesian equation by which we
can represent the different curves in question. If we refer the
tore to three axes of coordinates at right angles to one another
with the centre of the tore as origin, the axis of y being taken
to be the axis of revolution, and those of 0, x being perpen-
dicular to it in the plane bisecting the tore (making it a split-
ring), the equation of the tore is
(x 2 -f y 2 + z 2 + c 2 — a 2 ) 2 = 4 c 2 (z 2 + x 2 ),
206 SUCCESSORS OF THE GREAT GEOMETERS
where c, a have the same meaning as above. The different
sections parallel to the axis of revolution are obtained by
giving (say) z any value between and c + a. For the value
z — a the curve is the oval of Cassini which has the property
that, if r, r' be the distances of any point on the curve from
two fixed points as poles, W — const. For, if z — a, the equa-
tion becomes
(x 2 + y 2 + c 2 ) 2 = 4 <*x* + 4 c 2 a 2 ,
or {c — x 2 +y 2 } {c+x 2 + y 2 } = 4c 2 a 2 ;
and this is equivalent to rr = ±2ca if x, y are the coordinates
of any point on the curve referred to Ox, Oy as axes, where
is the middle point of the line (2 c in length) joining the two
poles, and Ox lies -along that line in either direction, while Oy
is perpendicular to it. Whether Perseus discussed this case
and arrived at the property in relation to the two poles is of
course quite uncertain.
Isoperimetric figures.
The subject of isoperimetric figures, that is to say, the com-
parison of the areas of figures having different shapes but the
same perimeter, was one which would naturally appeal to the
early Greek mathematicians. We gather from Proclus's notes
on Eucl. I. 36, 37 that those theorems, proving that parallelo-
grams or triangles on the same or equal bases and between
the same parallels are equal in area, appeared to the ordinary
person paradoxical because they meant that, by moving the
side opposite to the base in the parallelogram; or the yertex
of the triangle, to the right or left as far as we please, we may
increase the perimeter of the figure to any extent while keep-
ing the same area. Thus the perimeter in parallelograms or
triangles is in itself no criterion as to their area. Misconcep-
tion on this subject was rife among non-mathematicians.
Proclus tells us of describers of countries who inferred
the size of cities from their perimeters ; he mentions also
certain members of communistic societies in his own time who
cheated their fellow-members by giving them land of greater
perimeter but less area than the plots which they took
ISOPERIMETRIC FIGURES. ZENODORUS 207
themselves, so that, while they got a reputation for greater
honesty, they in fact took more than their share of the
produce. 1 Several remarks by ancient authors show the
prevalence of the same misconception. Thucydides estimates
the size of Sicily according to the time required for circum-
navigating it. 2 About 130 B.C. Polybius observed that there
were people who could not understand that camps of the same
periphery might have different capacities. 3 Quintilian has a
similar remark, and Cantor thinks he may have had in his
mind the calculations of Pliny, who compares the size of
different parts of the earth by adding their lengths to their
breadths. 4
Zenodorus wrote, at some date between (say) 200 B.C. and
A.D. 90, a treatise Trepi lo-ofxirpcov o-^fiaTcov, On isometric
figures. A number of propositions from it are preserved in
the commentary of Theon of Alexandria on Book I of
Ptolemy's Syntaxis ; and they are reproduced in Latin in the
third volume of Hultsch's edition of Pappus, for the purpose
of comparison with Pappus's own exposition of the same
propositions at the beginning of his Book V, where he appears
to have followed Zenodorus pretty closely while making some
changes in detail. 5 From the closeness with which the style
of Zenodorus follows that of Euclid and Archimedes we may
judge that his date was not much later than that of Archi-
medes, whom he mentions as the author of the proposition
(Measurement of a Circle, Prop. 1) that the area of a circle is
half that of the rectangle contained by the perimeter of the
circle and its radius. The important propositions proved by
Zenodorus and Pappus include the following: (1) Of all
regular 'polygons of equal perimeter, that is the greatest in
area which has the most angles. (2) A circle is greater than
any regular polygon of equal contour. (3) Of all polygons of
the same number of sides and equal perimeter the equilateral
and equiangular polygon is the greatest in area. Pappus
added the further proposition that Of all segments of a circle
having the same circumference the semicircle is the greatest in
1 Proclus on Eucl. I, p. 403. 5 sq. 2 Thuc. vi. 1.
3 Polybius, ix. 21. 4 Pliny, Hist. nat. vi. 208.
5 Pappus, v, p. 308 sq.
208 SUCCESSORS OF THE GREAT GEOMETERS
area. Zenodorus's treatise was not confined to propositions
about plane figures, but gave also the theorem that Of all
solid figures the surfaces of which are equal, the sphere is the
greatest in solid content.
We will briefly indicate Zenodorus's method of proof. To
begin with (1) ; let ABC, DEFhe equilateral and equiangular
polygons of the same perimeter, DEF having more angles
than ABC. Let G, H be the centres of the circumscribing
circles, GK, HL the perpendiculars from G, H to the sides
AB, DE, so that K , L bisect those sides.
AM
Since the perimeters are equal, AB > DE, and AK > DL.
Make KM equal to DL and join GM.
Since AB is the same fraction of the perimeter that the
angle A GB is of four right angles, and DE is the same fraction
of the same perimeter that the angle J)HE is of four right
angles, it follows that
AB:DE=lAGB:lDHE,
that is, AK : MK= LAGK-.L DHL.
But AK :MK > lAGK:l MGK
(this is easily proved in a lemma following by the usual
method of drawing an arc of a circle with G as centre and GM
as radius cutting GA and GK produced. The proposition is of.
course equivalent to tan a/ tan /S > x/fi, where \tt > a > /?).
Therefore Z MGK > Z DHL,
and consequently Z GMK < Z HDL.
Make the angle NMK equal to the angle HDL, so that MN
meets KG produced in N.
ZENODORUS 209
The triangles NMK, HDL are now equal in all respects, and
NK is equal to HL, so that GK < HL.
But the area of the polygon ABC is half the rectangle
contained by GK and the perimeter, while the area of the
polygon DEF is half the rectangle contained by HL and
the same perimeter. Therefore the area of the polygon DEF
is the greater.
(2) The proof that a circle is greater than any regular
polygon with the same perimeter is deduced immediately from
Archimedes's proposition that the area of a circle is equal
to the right-angled triangle with perpendicular side equal to
the radius and base equal to the perimeter of the circle ;
Zenodorus inserts a proof in extenso of Archimedes's pro-
position, with preliminary lemma. The perpendicular from
the centre of the circle circumscribing the polygon is easily
proved to be less than the radius of the given circle with
perimeter equal to that of the polygon ; whence the proposition
follows.
(3) The proof of this proposition depends on some pre-
liminary lemmas. The first proves that, if there be two
triangles on the same base and with the
same perimeter, one being isosceles and
the other scalene, the isosceles triangle
has the greater area. (Given the scalene
triangle BDC on the base BC, it is easy to
draw on BC as base the isosceles triangle
having the same perimeter. We have
only to take BH equal to ±(BD + DC),
bisect BC at E, and erect at E the per-
pendicular AE such that AE 2 = BH 2 -BE\)
Produce BA to F so that BA — AF, and join AD, DF.
Then BD + DF> BF, i.e. BA + AC, i.e. BD + DC, by hypo-
thesis; therefore DF > DC, whence in the triangles FAD,
CAD the angle FAD > the angle CAD.
Therefore Z FA D > \ Z FA C
> LBCA.
Make the angle FAG equal to the angle BC A or ABC, so
that AG is parallel to BC; let BD produced meet AG in G,
and join GG.
1523.2 P
210 SUCCESSORS OF THE GREAT GEOMETERS
Then
A ABC = A GBC
> ADBC.
The second lemma is to the effect that, given two isosceles
triangles not similar to one another, if we construct on the
same bases two triangles similar to one another such that the
sum of their perimeters is equal to the sum of the perimeters
of the first two triangles, then the sum of the areas of the
similar triangles is greater than the sum of the areas of
the non-similar triangles. (The easy construction of the
similar triangles is given in a separate lemma.)
Let the bases of the isosceles triangles, EB, BC\ be placed in
one straight line, BG being greater than EB.
Let ABC, DEB be the similar isosceles triangles, and FBG,
GEB the non-similar, the triangles being such that
BA + AC + ED +DB = BF+ FG+EG + GB.
Produce AF, GD to meet the bases in K, L. Then clearly
AK, GL bisect BG, EB at right angles at K, L.
Produce GL to H, making LH equal to GL.
Join HB and produce it to J\ r ; join HF.
Now, since the triangles A BG, DEB are similar, the angle
ABC is equal to the angle DEB or DBE.
Therefore Z NBG ( = Z HBE = Z GBE) > Z DBE or Z ABC ;
therefore the angle ABH f and a fortiori the angle FBH, is
less than two right angles, and HF meets BK in some point M.
ZENODORUS 211
Now, by hypothesis, DB + BA = GB + BF;
therefore DB + BA=HB + BF> HF.
By an easy lemma, since the triangles DEB, ABC are similar,
(DB + BAf = {DL + AKf + (BL + BK) 2
= (DL + AK)* + LK\
Therefore {DL + AKf + LK 2 > HF 2
>{GL + FK) 2 + LK 2 ,
whence DL + A K > GL + FK,
and it follows that AF > GD.
But BK > BL; therefore AF.BK > GD.BL.
Hence the ' hollow-angled (figure) ' (KoiXoycoviov) ABFC is
greater than the hollow-angled (figure) GEDB.
Adding A DEB + A BFG to each, we have h
ADEB + £ABC> AGEB+AFBC.
The above is the only case taken by Zenodorus. The proof
still holds if EB = BG, so that BK = BL. But it fails in the
case in which EB > BG and the vertex G of the triangle EB
belonging to the non-similar pair is still above D and not
below it (as F is below A in the preceding case). This was
no doubt the reason why Pappus gave a proof intended to
apply to all the cases without distinction. This proof is the
same as the above proof by Zenodorus up to the point where
it is proved that
DL + AK > GL + FK,
but there diverges. Unfortunately the text is bad, and gives
no sufficient indication of the course of the proof ; but it would
seem that Pappus used the relations
DL : GL = A DEB : A GEB,
AK : FK = A ABC: A FBC,
and AK 2 :DL 2 =A ABC: A DEB,
combined of course with the fact that GB + BF = DB + BA,
in order to prove the proposition that,
according as DL + AK > or < GL + FK,
ADEB + AABC> or < AGEB + AFBC.
p2
212 SUCCESSORS OF THE GREAT GEOMETERS
The proof of his proposition, whatever it was, Pappus
indicates that he will give later ; but in the text as we have it
the promise is not fulfilled.
Then follows the proof that the maximum polygon of given
perimeter is both equilateral and
equiangular.
(1) It is equilateral.
For, if not, let two sides of the
maximum polygon, as AB, BC, be
unequal. Join AC, and on iC as
base draw the isosceles triangle AFC
such that AF+ FC = AB + BC. The
area of the triangle AFC is then
greater than the area of the triangle ABC, and the area of
the whole polygon has been increased by the construc-
tion: which is impossible, as by hypothesis the area is a
maximum.
Similarly it can be proved that no other side is unequal
to any other.
(2) It is also equiangular.
For, if possible, let the maximum polygon ABCDE (which
we have proved to be equilateral)
have the angle at B greater than
the angle at D. Then BA C, DEC are
non-similar isosceles triangles. On
AC, CE as bases describe the two
isosceles triangles FAC, GEC similar
to one another which have the sum
of their perimeters equal to the
sum of the perimeters of BAG,
DEC. Then the sum of the areas of the two similar isosceles
triangles is greater than the sum of the areas of the triangles
BAC, DEC) the area of the polygon is therefore increased,
which is contrary to the hypothesis.
Hence no two angles of the polygon can be unequal.
The maximum polygon of given perimeter is therefore both
equilateral and equiangular.
Dealing with the sphere in relation to other solids having
ZENODORUS. HYPSICLES 213
their surfaces equal to that of the sphere, Zenodorus confined
himself to proving (1) that the sphere is greater if the other
solid with surface equal to that of the sphere is a solid formed
by the revolution of a regular polygon about a diameter
bisecting it as in Archimedes, On the Sphere and Cylinder,
Book I, and (2) that the sphere is greater than any of
the regular solids having its surface equal to that of the
sphere.
Pappus's treatment of the subject is more complete in that
he proves that the sphere is greater than the cone or cylinder
the surface of which is equal to that of the sphere, and further
that of the five regular solids which have the same surface
that which has more faces is the greater. 1
Hypsicles (second half of second century B.C.) has already
been mentioned (vol. i, pp. 419-20) as the author of the con-
tinuation of the Elements known as Book XIV. He is quoted
by Diophantus as having given a definition of a polygonal
number as follows :
' If there are as many numbers as we please beginning from
1 and increasing by the same common difference, then, when
the common difference is 1, the sum of all the numbers is
a triangular number; when 2, a square; when 3, a pentagonal
number [and so on]. And the number of angles is called
after the number which exceeds the common difference by 2,
and the side after the number of terms including 1.'
This definition amounts to saying that the nth. a-gonal num-
ber (1 counting as the first) is \n { 2 + {n— 1) (a — 2) }. If, as is
probable, Hypsicles wrote a treatise on polygonal numbers, it
has not survived. On the other hand, the 'AvacpopiKos (Ascen-
siones) known by his name has survived in Greek as well as in
Arabic, and has been edited with translation. 2 True, the
treatise (if it really be by Hypsicles, and not a clumsy effort
by a beginner working from an original by Hypsicles)
does no credit to its author; but it is in some respects
interesting, and in particular because it is the first Greek
1 Pappus, v, Props. 19, 38-56.
2 Manitius, Des Hypsikles Schrift Anaphorikos, Dresden, Lehmannsche
Buchdruckerei, 1888.
214 SUCCESSORS OF THE GREAT GEOMETERS
work in which we find the division of the ecliptic circle into
360 ' parts ' or degrees. The author says, after the preliminary-
propositions,
'The circle of the zodiac having been divided into 360 equal
circumferences (arcs), let each of the latter be called a degree
in space (fiolpa tottiktj, 'local' or 'spatial part'). And simi-
larly, supposing that the time in which the zodiac circle
returns to any position it has left is divided into 360 equal
times, let each of these be called a degree in time (/ioipa
XpOVLKTj)'
From the word KaXeiaOco (' let it be called ') we may perhaps
infer that the terms were new in Greece. This brings us to
the question of the origin of the division (1) of the circle of
the zodiac, (2) of the circle in general, into 360 parts. On this
question innumerable suggestions have beerf made. With
reference to (1) it was suggested as long ago as 1788 (by For-
maleoni) that the division was meant to correspond to the
number of days in the year. Another suggestion is that it
would early be discovered that, in the case of any circle the
inscribed hexagon dividing the circumference into six parts
has each of its sides equal to the radius, and that this would
naturally lead to the circle being regularly divided into six
parts ; after this, the very ancient sexagesimal system would
naturally come into operation and each of the parts would be
divided into 60 subdivisions, giving 360 of these for the whole
circle. Again, there is an explanation which is not even
geometrical, namely that in the Babylonian numeral system,
which combined the use of 6 and 10 as bases, the numbers 6,
60, 360, 3600 were fundamental round numbers, and these
numbers were transferred from arithmetic to the heavens.
The obvious objection to the first of these explanations
(referring the 360 to the number of days in the solar year) is
that the Babylonians were well acquainted, as far back as the
monuments go, with 365-2 as the number of days in the year.
A variant of the hexagon- theory is the suggestion that a
natural angle to be discovered, and to serve as a measure of
others, is the angle of an equilateral triangle, found by draw-
ing a star # like a six-spoked wheel without any circle. If
the base of a sundial was so divided into six angles, it would be
HYPSICLES 215
natural to divide each of the sixth parts into either 10 or 60
parts; the former division would account for the attested
division of the clay into 60 hours, while the latter division on
the sexagesimal system would give the 360 time-degrees (each
of 4 minutes) making up the day of 24 hours. The purely
arithmetical explanation is defective in that the series of
numbers for which the Babylonians had special names is not
60, 360, 3600 but 60 (Soss), 600 (Ner), and 3600 or 60 2 (Sar).
On the whole, after all that has been said, I know of no
better suggestion than that of Tannery. 1 It is certain that
both the division of the ecliptic into 360 degrees and that of
the wxOrjpepw into 360 time-degrees were adopted by the
Greeks from Babylon. Now the earliest division of the
ecliptic was into 12 parts, the signs, and the question is, how
were the signs subdivided ? Tannery observes that, accord-
ing to the cuneiform inscriptions, as well as the testimony of
Greek authors, the sign was divided into parts one of which
(dargatu) was double of the other (murra n), the former being
l/30th, the other (called stadium by Manilius) l/60th, of the
sign ; the former division would give 360 parts, the latter 720
parts for the whole circle. The latter division was more
natural, in view of the long-established system of sexagesimal
fractions; it also had the advantage of corresponding toler-
ably closely to the apparent diameter of the sun in comparison
with the circumference of the sun's apparent circle. But, on
the other hand, the double fraction, the l/30th, was contained
in the circle of the zodiac approximately the same number of
times as there are days in the year, and consequently corre-
sponded nearly to the distance described by the sun along the
zodiac in one day. It would seem that this advantage was
sufficient to turn the scale in favour of dividing each sign of
the zodiac into 30 parts, giving 360 parts for the whole
circle. While the Chaldaeans thus divided the ecliptic into
360 parts, it does not appear that they applied the same divi-
sion to the equator or any other circle. They measured angles
in general by ells, an ell representing 2°, so that the complete
circle contained 180, not 360, parts, which they called ells.
The explanation may perhaps be that the Chaldaeans divided
1 Tannery, 'La coudee astronomique et les anciennes divisions du
cercle ' (Memo ires scientifiques, ii, pp. 256-68).
216 SUCCESSORS OF THE GREAT GEOMETERS
the diameter of. the circle into 60 ells in accordance with their
usual sexagesimal division, and then came to divide the cir-
cumference into 180 such ells on the ground that the circum-
ference is roughly three times the diameter. The measure-
ment in ells and dactyli (of which there were 24 to the ell)
survives in Hipparchus (On the Phaenomena of Eudoxus and
Aratus), and some measurements in terms of the same units
are given by Ptolemy. It was Hipparchus who first divided
the circle in general into 360 parts or degrees, and the
introduction of this division coincides with his invention of
trigonometry.
The contents of Hypsicles's tract need not detain us long.
The problem is : If we know the ratio which the length of the
longest day bears to the length of the shortest day at any
given place, to find how many time-degrees it takes any given
sign to rise ; and, after this has been found, the author finds
what length of time it takes any given degree in any sign to
rise, i.e. the interval between the rising of one degree-point on
the ecliptic and that of the next following. It is explained
that the longest day is the time during which one half of the
zodiac (Cancer, Leo, Virgo, Libra, Scorpio, Sagittarius) rises,
and the shortest day the time during which the other half
(Capricornus, Aquarius, Pisces, Aries, Taurus, Gemini) rises.
Now at Alexandria the longest day is to the shortest as 7
to 5; the longest therefore contains 210 'time-degrees', the
shortest 150. The two quadrants Cancer-Virgo and Libra-
Sagittarius take the same time to rise, namely 105 time-
degrees, and the two quadrants Capricornus-Pisces and Aries-
Gemini each take the same time, namely 75 time-degrees.
It is further assumed that the times taken by Virgo, Leo,
Cancer, Gemini, Taurus, Aries are in descending arithmetical
progression, while the times taken by Libra, Scorpio, Sagit-
tarius, Capricornus, Aquarius, Pisces continue the same de-
scending arithmetical series. The following lemmas are used
and proved :
I. If a 1? a 2 ...a n , a n + J > a w+2 ... a 2w is a descending arithmeti-
cal progression of 2 n terms with 8 ( = a x — <x 2 = a 2 — a 3 = . . .)
as common difference,
ai + a*+...+a tt -(a n+1 + a n+2 +...+a 2n ) = n 2 8.
HYPSICLES 217
II. If Oj, a 2 ... a n ... a 2 «-i i s a descending arithmetical pro-
gression of 2n—l terms with 8 as common difference and a n
is the middle term, then
a 1 + « 2 +...+a 2n _ 1 = (27i-l)a n .
III. If a 1} a 2 ...a n , a n+1 ...a 2n is a descending arithmetical
progression of 2n terms, then
a x + a 2 + . . . + a 2 n = w (a x + a 2 n ) = n (a 2 + a 2 n _ 2 ) = . . .
= ^(a w + a n+1 ).
Now let J., 5, G be the descending series the sum of which
is 105, and D, E, F the next three terms in the same series
the sum of which is 75, the common difference being 8; we
then have, by (I),
A + B + G-(D + E+F) = 98, or 30 = 98,
and 8= 3|.
Next, by (II), A + B + C =3B, or 3B = 105, and B = 35 ;
therefore ^4, i?, (7, 2), i?, .Fare equal to 38§, 35, 31 §, 28-J, 25,
21§ time-degrees respectively, which the author of the tract
expresses in time-degrees and minutes as 38*20', 35*, 31*40',
28* 20', 25*, 21* 40'. We have now to carry through the same
procedure for each degree in each sign. If the difference
between the times taken to rise by one sign and the next
is 3* 20', what is the difference for each of the 30 degrees in
the sign 1 We have here 30 terms followed by 30 other terms
of the same descending arithmetical progression, and the
formula (I) gives 3* . 20' = (30) 2 <i, where d is the common
difference ; therefore d = ^-g x 3* . 20'= 0* 0' 13" 20'". Lastly,
take the sign corresponding to 21* 40'. This is the sum of
a descending arithmetical progression of 30 terms a lt a 2 ... a M
with common difference 0* 0' 13" 20'". Therefore, by (III),
21* 40' = \b(a 1 + a m ), whence a 1 + a 50 — 1* 26' 40". Now,
since there are 30 terms a 17 a 2 ... a 30 , we have
a x -a^ = 29d = 0* 6' 26" 40"'.
It follows that a. 30 = 0* 40' 6" 40'" and a x = 0* 46' 33" 20'",
218 SUCCESSORS OF THE GREAT GEOMETERS
and from these and the common difference 0* O'lS'^O'" all
the times corresponding to all the degrees in the circle can be
found.
The procedure was probably, as Tannery thinks, taken
direct from the Babylonians, who would no doubt use it for
the purpose of enabling the time to be determined at any
hour of the night. Another view is that the object was
astrological rather than astronomical (Manitius). In either
case the method was exceedingly rough, and the assumed
increases and decreases in the times of the risings of the signs
in arithmetical progression are not in accordance with the
facts. The book could only have been written before the in-
vention of trigonometry by Hipparchus, for the problem of
finding the times of rising of the signs is really one of
spherical trigonometry, and these times were actually cal-
culated by Hipparchus and Ptolemy by means of tables of
chords.
Dionysodorus is known in the first place as the author of
a solution of the cubic equation subsidiary to the problem of
Archimedes, On the Sjrfiere and Cylinder, II. 4, To cut a given
sphere by a plane so that the volumes of the segments have to
one another a given ratio (see above, p. 46). Up to recently
this Dionysodorus was supposed to be Dionysodorus of Amisene
in Pontus, whom Suidas describes as ' a mathematician worthy
of mention in the field of education '. But we now learn from
a fragment of the Herculaneum Roll, No. 1044, that ' Philonides
was a pupil, first of Eudemus, and afterwards of Dionysodorus,
the son of Dionysodorus the Caunian'. Now Eudemus is
evidently Eudemus of Pergamum to whom Apollonius dedi-
cated the first two Books of his Conies, and Apollonius actually
asks him to show Book II to Philonides. In another frag-
ment Philonides is said to have published some lectures by
Dionysodorus. Hence our Dionysodorus may be Dionysodorus
of Caunus and a contemporary of Apollonius, or very little
later. 1 A Dionysodorus is also mentioned by Heron 2 as the
author of a tract On the ^ire (or tore) in which he proved
that, if d be the diameter of the revolving circle which
1 W. Schmidt in Bibliotheca mathematica, iv 3 , pp. 321-5.
2 Heron, Metrica, ii. 13, p. 128. 3.
DIONYSODORUS 219
generates the tore, and c the distance of its centre from the
axis of revolution,
(volume of tore) \ttc 2 .d = ^7rd 2 :^cd i
that is, (volume of tore) = \tt 2 . cd 2 ,
which is of course the product of the area of the generating
circle and the length of the path of its centre of gravity. The
form in which the result is stated, namely that the tore is to
the cylinder with height d and radius c as the generating
circle of the tore is to half the parallelogram cd, indicates
quite clearly that Dionysodorus proved his result by the same
procedure as that employed by Archimedes in the Method and
in the book On Conoids and Sjjheroids ; and indeed the proof
on Archimedean lines is not difficult.
Before passing to the mathematicians who are identified
with the discovery and development of trigonometry, it will
be convenient, I think, to dispose of two more mathematicians
belonging to the last century B.C., although this involves
a slight departure from chronological order ; I mean Posidonius
and Geminus.
Posidonius, a Stoic, the teacher of Cicero, is known as
Posidonius of Apamea (where he was born) or of Rhodes
(where he taught) ; his date may be taken as approximately
135-51 B.C. In pure mathematics he is mainly quoted as the
author of certain definitions, or for views on technical terms,
e.g. ' theorem ' and ' problem ', and subjects belonging to ele-
mentary geometry. More important were his contributions
to mathematical geography and astronomy. He gave his
great work on geography the title On the Ocean, using the
word which had always had such a fascination for the Greeks ;
its contents are known to us through the copious quotations
from it in Strabo ; it dealt with physical as well as mathe-
matical geography, the zones, the tides and their connexion
with the moon, ethnography and all sorts of observations made
during extensive travels. His astronomical book bore the
title Meteorologica or nepl uereoopcov, and, while Geminus
wrote a commentary on or exposition of this work, we may
assign to it a number of views quoted from Posidonius in
220 SUCCESSORS OF THE GREAT GEOMETERS
Cleomedes's work De motu circulari corporum caelestium.
Posidonius also wrote a separate tract on the size of the sun.
The two things which are sufficiently important to deserve
mention here are (1) Posidonius's measurement of the circum-
ference of the earth, (2) his hypothesis as to the distance and
size of the sun. *
(1) He estimated the circumference of the earth in this
way. He assumed (according to Cleomedes l ) that, whereas
the star Canopus, invisible in Greece, was just seen to graze the
horizon at Rhodes, rising and setting again immediately, the
meridian altitude of the same star at Alexandria was ' a fourth
part of a sign, that is, one forty-eighth part of the zodiac
circle' (= 7-|°) ; and he observed that the distance between
the two places (supposed to lie on the same meridian) ' was
considered to be 5,000 stades'. The circumference of the
earth was thus made out to be 240,000 stades. Unfortunately
the estimate of the difference of latitude, 7^°, was very far
from correct, the true difference being 5^° only ; moreover
the estimate of 5,000 stades for the distance was incorrect*
being only the maximum estimate put upon it by mariners,
while some put it at 4,000 and Eratosthenes, by observations
of the shadows of gnomons, found it to be 3,750 stades only.
Strabo, on the other hand, says that Posidonius favoured ■ the
latest of the measurements which gave the smallest dimen-
sions to the earth, namely about 180,000 stades'. 2 This is
evidently 48 times 3,750, so that Posidonius combined Erato-
sthenes's figure of 3,750 stades with the incorrect estimate
of 7\° for the difference of latitude, although Eratosthenes
presumably obtained the figure of 3,750 stades from his own
estimate (250,000 or 252,000) of the circumference of the earth
combined with an estimate of the difference of latitude which
was about 5-1° and therefore near the truth.
5
(2) Cleomedes 3 tells us that Posidonius supposed the circle
in which the sun apparently moves round the earth to be
10,000 times the size of a circular section of the earth through
its centre, and that with this assumption he combined the
1 Cleomedes, Be motu circulari, i. 10, pp. 92-4.
2 Strabo, ii. c. 95.
3 Cleomedes, op. cit. ii. 1, pp. 144-6, p. 98. 1-5.
POSIDONIUS 221
statement of Eratosthenes (based apparently upon hearsay)
that at Syene, which is under the summer tropic, and
throughout a circle round it of 300 stades in diameter, the
upright gnomon throws no shadow at noon. It follows from
this that the diameter of the sun occupies a portion of the
sun's circle 3,000,000 stades in length ; in other words, the
diameter of the sun is 3,000,000 stades. The assumption that
the sun's circle is 10,000 times as large as a great circle of the
earth was presumably taken from Archimedes, who had proved
in the Sand-reckoner that the diameter of the sun's orbit is
less than 10,000 times that of the earth; Posidonius in fact
took the maximum value to be the true value ; but his esti-
mate of the sun's size is far nearer the truth than the estimates
of Aristarchus, Hipparchus, and Ptolemy. Expressed in terms
of the mean diameter of the earth, the estimates of these
astronomers give for the diameter of the sun the figures 6|,
12§, and 5^ respectively; Posidonius's estimate gives 39J, the
true figure being 108-9.
In elementary geometry Posidonius is credited by Proclus
with certain definitions. He defined ' figure ' as ' confining
limit' (wepa? crvy K\dov) 1 and 'parallels' as 'those lines which,
being in one plane, neither converge nor diverge, but have all
the perpendiculars equal which are drawn from the points of
one line to the other'. 2 (Both these definitions are included
in the Definitions of Heron.) He also distinguished seven
species of quadrilaterals, and had views on the distinction
between theorem and problem. Another indication of his
interest in the fundamentals of elementary geometry is the
fact 3 that he wrote a separate work in refutation of the
Epicurean Zeno of Sidon, who had objected to the very begin-
nings of the Elements on the ground that they contained un-
proved assumptions. Thus, said Zeno, even Eucl.1. 1 requires it
to be admitted that ' two straight lines cannot have a common
segment ' ; and, as regards the ' proof ' of this fact deduced
from the bisection of a circle by its diameter, he would object
that it has to be assumed that two arcs of circles cannot have
a common part. Zeno argued generally that, even if we
admit the fundamental principles of geometry, the deductions
1 Proclus on Eucl. I, p. 143. 8. 2 lb., p. 176. 6-10.
3 lb., pp. 199. 14-200. 3.
222 SUCCESSORS OF THE GREAT GEOMETERS
from them cannot be proved without the admission of some-
thing else as well which has not been included in the said
principles, and he intended by means of these criticisms to
destroy the whole of geometry. 1 We can understand, there-
fore, that the tract of Posidonius was a serious work.
A definition of the centre of gravity by one ' Posidonius a
Stoic ' is quoted in Heron's Mechanics, but, as the writer goes
on to say that Archimedes introduced a further distinction, we
may fairly assume that the Posidonius in question is not
Posidonius of Rhodes, but another, perhaps Posidonius of
Alexandria, a pupil of Zeno of Cittium in the third cen-
tury B.C.
We now come to Geminus, a very important authority on
many questions belonging to the history of mathematics, as is
shown by the numerous quotations from him in Proclus's
Commentary on Euclid, Book I. His date and birthplace are
uncertain, and the discussions on the subject now form a whole
literature for which reference must be made to Manitius's
edition of the so-called Gemini element a astronomiae (Teubner,
1898) and the article 'Geminus' in Pauly-Wissowa's Real-
Encyclopadie. The doubts begin with his name. Petau, who
included the treatise mentioned in his Uranologion (Paris,
1630), took it to be the Latin Geminus. Manitius, the latest
editor, satisfied himself that it was Geminus, a Greek name,
judging from the fact that it consistently appears with the
properispomenon accent in Greek {Teiuvos), while it is also
found in inscriptions with the spelling Pe/xe^o?; Manitius
suggests the derivation from ye/z, as 'EpyTvos from epy, and
'A\e£tvos from aAe£ ; he compares also the unmistakably
Greek names 'IktIvos, Kparivos. Now, however, we are told
(by Tittel) that the name is, after all, the Latin Ge'mmus,
that re/jLiuo? came to be so written through false analogy
with 'AXegivos, &c, and that Te^elvos, if the reading is
correct, is also wrongly formed on the model of Avrccvelvo^,
Aypnnrdva. The occurrence of a Latin name in a centre
of Greek culture need not surprise us, since Romans settled in
such centres in large numbers during the last century B.C.
Geminus, however, in spite of his name, was thoroughly Greek.
1 Proclus on Eucl. I, pp. 214. 18-215. 13, p. 216. 10-19, p. 217. 10-23.
GEMINUS 223
An upper limit for his date is furnished by the fact that he
wrote a commentary on or exposition of Posidonius's work
nepl fi€T€dopcou ; on the other hand, Alexander Aphrodisiensis
(about a.d. 210) quotes an important passage from an 'epitome'
of this egrjyrja-is by Geminus. The view most generally
accepted is that he was a Stoic philosopher, born probably
in the island of Rhodes, and a pupil of Posidonius, and that
he wrote about 73-67 B.C.
Of Geminus's works that which has most interest for us
is a comprehensive work on mathematics. Proclus, though
he makes great use of it, does not mention its title, unless
indeed, in the passage where, after quoting from Geminus
a classification of lines which never meet, he says ' these
remarks I have selected from the (piXoKaXta of Geminus', 1
the word (piXoKaXia is a title or an alternative title. Pappus,
however, quotes a work of Geminus ' on the classification of
the mathematics' (kv tS> irepl rfjs tqov fjLaOrjfidrcoy radons),
while Eutocius quotes from ' the sixth book of the doctrine of
the mathematics ' (ev tS> Zkto) rr]s tS>v fiaOrjfiaTcoy Oeoopias).
The former title corresponds well enough to the long extract
on the division of the mathematical sciences into arithmetic,
geometry, mechanics, astronomy, optics, geodesy, canonic
(musical harmony) and logistic which Proclus gives in his
first prologue, and also to the fragments contained in the
Anonymi variae collectiones published by Hultsch in his
edition of Heron ; but it does not suit most of the other
passages borrowed by Proclus. The correct title was most
probably that given by Eutocius, The Doctrine, or Theory,
of the Mathematics) and Pappus probably refers to one
particular section of the work, say the first Book. If the
sixth Book treated of conies, as we may conclude from
Eutocius's reference, there must have been more Books to
follow; for Proclus has preserved us details about higher
curves, which must have come later. If again Geminus
finished his work and wrote with the same fullness about the
other branches of mathematics as he did about geometry,
there must have been a considerable number of Books
altogether. It seems to have been designed to give a com-
plete view of the whole science of mathematics, and in fact
1 Proclus on Eucl. I, p. 177. 24.
224 SUCCESSORS OF THE GREAT GEOMETERS
to have been a sort of encyclopaedia of the subject. The
quotations of Proclus from Geminus's work do not stand
alone; we have other collections of extracts, some more and
some less extensive, and showing varieties of tradition accord-
ing to the channel through which they came down. The
scholia to Euclid's Elements, Book I, contain a considerable
part of the commentary on the Definitions of Book I, and are
valuable in that they give Geminus pure and simple, whereas
Proclus includes extracts from other authors. Extracts from
Geminus of considerable length are included in the Arabic
commentary by an-Nairizi (about A.D. 900) who got them
through the medium of Greek commentaries on Euclid,
especially that of Simplicius. It does not appear to be
doubted any longer that ' Aganis ' in an-Nairizi is really
Geminus ; this is inferred from the close agreement between
an-Nairizi's quotations from ' Aganis ' and the correspond-
ing passages in Proclus ; the difficulty caused by the fact
that Simplicius calls Aganis ' socius noster ' is met by the
suggestion that the particular word socius is either the
result of the double translation from the Greek or means
nothing more, in the mouth of Simplicius, than ' colleague '
in the sense of a worker in the same field, or ' authority \
A few extracts again are included in the Anonymi variae
collectiones in Hultsch's Heron, Nos. 5-14 give definitions of
geometry, logistic, geodesy and their subject-matter, remarks
on bodies as continuous magnitudes, the three dimensions as
' principles ' of geometry, the purpose of geometry, and lastly
on optics, with its subdivisions, optics proper, Gatoptriea and
o-KT}uoypa(f>LKrj, scene-painting (a sort of perspective), with some
fundamental principles of optics, e.g. that all light travels
along straight lines (which are broken in the cases of reflection
and refraction), and the division between optics and natural
philosophy (the theory of light), it being the province of the
latter to investigate (what is a matter of indifference to optics)
whether (1) visual rays issue from the eye, (2) images proceed
from the object and impinge on the eye, or (3) the intervening
air is aligned or compacted with the beam-like breath or
emanation from the eye.
Nos. 80-6 again in the same collection give the Peripatetic
explanation of the name mathematics, adding that the term
GEMINUS 225
was applied by the early Pythagoreans more particularly
to geometry and arithmetic, sciences which deal with the pure,
the eternal and the unchangeable, but was extended by later
writers to cover what we call ' mixed ' or applied mathematics,
which, though theoretical, has to do with sensible objects, e.g.
astronomy and optics. Other extracts from Geminus are found
in extant manuscripts in connexion with Damianus's treatise
on optics (published by R. Schone, Berlin, 1897). The defini-
tions of logistic and geometry also appear, but with decided
differences, in the scholia to Plato's Charmides 165 e. Lastly,
isolated extracts appear in Eutocius, (1) a remark reproduced
in the commentary on Archimedes's Plane Equilibriums to
the effect that Archimedes in that work gave the name of
postulates to what are really axioms, (2) the statement that
before Apollonius's time the conies were produced by cutting
different cones (right-angled, acute-angled, and obtuse-angled)
by sections perpendicular in each case to a generator. 1
The object of Geminus's work was evidently the examina-
tion of the first principles, the logical building up of mathe-
matics on the basis of those admitted principles, and the
defence of the whole structure against the criticisms of
the enemies of the science, the Epicureans and Sceptics, some
of whom questioned the unproved principles, and others the
logical validity of the deductions from them. Thus in
geometry Geminus dealt first with the principles or hypotheses
(dp^ai, vTroBecreis) and then with the logical deductions, the
theorems and problems (rot jiera ras dp\d?). The distinction
is between the things which must be taken for granted but
are incapable of proof and the things which must not be
assumed but are matter for demonstration. The principles
consisting of definitions, postulates, and axioms, Geminus
subjected them severally to a critical examination from this
point of view, distinguishing carefully between postulates and
axioms, and discussing the legitimacy or otherwise of those
formulated by Euclid in each class. In his notes on the defini-
tions Geminus treated them historically, giving the various
alternative definitions which had been suggested for each
fundamental concept such as ( line ', ' surface ', ' figure ', 'body',
' angle ', &c, and frequently adding instructive classifications
1 Eutocius, Comm. on Apollonius's Conies, ad init,
1523.2 Q
225 SUCCESSORS OF THE GREAT GEOMETERS
•of the different species of the thing defined. Thus in the
case of ' lines ' (which include curves) he distinguishes, first,
the composite (e.g. a broken line forming an angle) and the
incomposite. The incomposite are subdivided into those
' forming a figure ' (o"x^ fiaTonoiovo-ai) or determinate (e.g.
circle, ellipse, cissoid) and those not forming a figure, inde-
terminate and extending without limit (e. g. straight line,
parabola, hyperbola, conchoid). In a second classification
incomposite lines are divided into (1) ' simple ', namely the circle
and straight line, the one ' making a figure ', the other extend-
ing without limit, and (2) 'mixed'. 'Mixed' lines again are
divided into (a) 'lines in planes', one kind being a line meet-
ing itself (e.g. the cissoid) and another a line extending
without limit, and (b) ' lines on solids ', subdivided into lines
formed by sections (e.g. conic sections, spiric curves) and
'lines round solids' (e.g. a helix round a cylinder, sphere, or
cone, the first of which is uniform, homoeomeric, alike in all
its parts, while the others are non-uniform). Geminus gave
a corresponding division of surfaces into simple and mixed,
the former being plane surfaces and spheres, while examples
of the latter are the tore or anchor-ring (though formed by
the revolution of a circle about an axis) and the conicoids of
revolution (the right-angled conoid, the obtuse-angled conoid,
and the two spheroids, formed by the revolution of a para-
bola, a hyperbola, and an ellipse respectively about their
axes). He observes that, while there are three homoeomeric
or uniform ' lines ' (the straight line, the circle, and the
cylindrical helix), there are only two homoeomeric surfaces,
the plane and the sphere. Other classifications are those of
' angles ' (according to the nature of the two lines or curves
which form them) and of figures and plane figures.
When Proclus gives definitions, &c, by Posidonius, it is
evident that he obtained them from Geminus's work. Such
are Posidonius's definitions of ' figure ' and ' parallels ', and his
division of quadrilaterals into seven kinds. We may assume
further that, even where Geminus did not mention the name
of Posidonius, he was, at all events so far as the philosophy of
mathematics was concerned, expressing views which were
mainly those of his master.
GEMINUS 227
Attempt to prove the Parallel-Postulate.
Geminus devoted much attention to the distinction between
postulates and axioms, giving the views of earlier philoso-
phers and mathematicians (Aristotle, Archimedes, Euclid,
Apollonius, the Stoics) on the subject as well as his own. It
was important in view of the attacks of the Epicureans and
Sceptics on mathematics, for (as Geminus says) it is as futile
to attempt to prove the indemonstrable (as Apollonius did
when he tried to prove the axioms) as it is incorrect to assume
what really requires proof, ' as Euclid did in the fourth postu-
late [that all right angles are equal] and in the fifth postulate
[the parallel-postulate] '}
The fifth postulate was the special stumbling-block.
Geminus observed that the converse is actually proved by
Euclid in I. 17; also that it is conclusively proved that an
angle equal to a right angle is not necessarily itself a right
angle (e.g. the ' angle ' between the circumferences of two semi-
circles on two equal straight lines with a common extremity
and at right angles to one another) ; we cannot therefore admit
that the converses are incapable of demonstration. 2 And
' we have learned from the very pioneers of this science not to
have regard to mere plausible imaginings when it is a ques-
tion of the reasonings to be included in our geometrical
doctrine. As Aristotle says, it is as justifiable to ask scien-
tific proofs from a rhetorician as to accept mere plausibilities
from a geometer ... So in this case (that of the parallel-
postulate) the fact that, when the right angles are lessened, the
straight lines converge is true and necessary ; but the state-
ment that, since they converge more and more as they are
produced, they will sometime meet is plausible but not neces-
sary, in the absence of some argument showing that this is
true in the case of straight lines. For the fact that some lines
exist which approach indefinitely but yet remain non-secant
(dcrvfi7rTCQToi), although it seems improbable and paradoxical,
is nevertheless true and fully ascertained with reference to
other species of lines [the hyperbola and its asymptote and
the conchoid and its asymptote, as Geminus says elsewhere].
May not then the same thing be possible in the case of
1 Proclus on Eucl. I, pp. 178-82. 4; 183. 14-184. 10.
2 lb., pp. 183. 26-184. 5.
Q 2
228 SUCCESSORS OF THE GREAT GEOMETERS
straight lines which happens in the case of the lines referred
to? Indeed, until the statement in the postulate is clinched
by proof, the facts shown in the case of the other lines may
direct our imagination the opposite way. And, though the
controversial arguments against the meeting of the straight
lines should contain much that is surprising, is there not all
the more reason why we should expel from our body of
doctrine this merely plausible and unreasoned (hypothesis) ?
It is clear from this that we must seek a proof of the present
theorem, and that it is alien to the special character of
postulates.' l
Much of this might have been written by a modern
geometer. Geminus's attempted remedy was to substitute
a definition of parallels like that of Posidonius, based on the
notion of eqvAdistance. An-Nairizi gives the definition as
follows : ' Parallel straight lines are straight lines situated in
the same plane and such that the distance between them, if
they are produced without limit in both directions at the same
time, is everywhere the same ', to which Geminus adds the
statement that the said distance is the shortest straight line
that can be drawn between them. Starting from this,
Geminus proved to his own satisfaction the propositions of
Euclid regarding parallels and finally the parallel-postulate.
He first gave the propositions (1) that the 'distance ' between
the two lines as defined is perpendicular to both, and (2) that,
if a straight line is perpendicular to each of two straight lines
and meets both, the two straight lines are parallel, and the
' distance ' is the intercept on the perpendicular (proved by
reductio ad absurdum). Next come (3) Euclid's propositions
I. 27, 28 that, if two lines are parallel, the alternate angles
made by any transversal are equal, &c. (easily proved by
drawing the two equal ' distances ' through the points of
intersection with the transversal), and (4) Eucl. I. 29, the con-
verse of I. 28, which is proved lyy reductio ad absurdum, by
means of (2) and (3). Geminus still needs Eucl. I. 30, 31
(about parallels) and I. 33, 34 (the first two propositions
relating to parallelograms) for his final proof of the postulate,
which is to the following effect.
Let A B, CD be two straight lines met by the straight line
1 Proclus on Eucl. I, pp. 192. 5-193. 3.
GEMINUS
229
EF, and let the interior angles BEF, EFD be together less
than two right angles.
Take any point H on FD and draw HK parallel to AB
meeting EF in K. Then, if we bisect EF at L, LF at M, MF
at X, and so on, we shall at last have a length, as FN, less
than FK. Draw FG, NOP parallel to AB. Produce FO to Q,
and let i^Q be the same multiple of FO that FE is of i^iY ;
then shall AB, CD meet in Q.
Let $ be the middle point of FQ and R the middle point of
FS. Draw through R, S, Q respectively the straight lines
RPG, STU, QV parallel to EF. Join MR, LS and produce
them to T, V Produce FG to U.
Then, in the triangles FON, ROP, two angles are equal
respectively, the vertically opposite angles FON, ROP and
the alternate angles NFO, PRO ; and FO = OR ; therefore
RP = FK
And FN, PG in the parallelogram FNPG are equal ; there-
fore RG = 2FN= FM (whence MR is parallel to FG or AB)
Similarly we prove that SU = 2 FM = FL, and LS is
parallel to FG or AB.
Lastly, by the triangles FLS, QVS, in which the sides FS,
SQ are equal and two angles are respectively equal, Q V = FL.
Therefore QV = LE.
Since then EL, QV are equal and parallel, so are EQ, LV,
and (says Geminus) it follows that AB passes through Q.
230 SUCCESSORS OF THE GREAT GEOMETERS
What follows is actually that both EQ and A B (of EB)
are parallel to LV, and Geminus assumes that EQ, AB
are coincident (in other words, that through a given point
only one parallel can be drawn to a given straight line, an
assumption known as PJayfair's Axiom, though it is actually
stated in Proclus on Eucl. I. 31).
The proof therefore, apparently ingenious as it is, breaks
down. Indeed the method is unsound from the beginning,
since (as Saccheri pointed out), before even the definition of
parallels by Geminus can be used, it has to be proved that
' the geometrical locus of points equidistant from* a straight
line is a straight line ', and this cannot be proved without a
postulate. But the attempt is interesting as the first which
has come down to us, although there must have. been many
others by geometers earlier than Geminus.
Coming now to the things which follow from the principles
(rd fxtTa ras dp\ds), we gather from Proclus that Geminus
carefully discussed such generalities as the nature of elements,
the different views which had been held of the distinction
between theorems and problems, the nature and significance
of Siopio-jioL (conditions and limits of possibility), the meaning
of ( porism ' in the sense in which Euclid used the word in his
Porisms as distinct from its other meaning of ' corollary ', the
different sorts of problems and theorems, the two varieties of
converses (complete and partial), topical or locus-theorems,
with the classification of loci. He discussed also philosophical
questions, e.g. the question whether a line is made up of
indivisible parts (e£ djxepcov), which came up in connexion
with Eucl. I. 10 (the bisection of a straight line).
The book was evidently not less exhaustive as regards
higher geometry. Not only did Geminus mention the spiric
curves, conchoids and cissoids in his classification of curves ;
he showed how they were obtained, and gave proofs, presum-
ably of their principal properties. Similarly he gave the
proof that there are three homoeomeric or uniform lines or
curves, the straight line, the circle and the cylindrical helix.
The proof of ' uniformity ' (the property that any portion of
the line or curve will coincide with any other portion of the
same length) was preceded by a proof that, if two straight
lines be drawn from any point to meet a uniform line or curve
GEMINUS 231
and make equal angles with it, the straight lines are equal. 1
As Apollonius wrote on the cylindrical helix and proved the
fact of its uniformity, we may fairly assume that Geminus
was here drawing upon Apollonius.
Enough has been said to show how invaluable a source of
information Geminus's work furnished to Proclus and all
writers on the history of mathematics who had access to it.
In astronomy we know that Geminus wrote an egrjyrjo-is of
Posidonius's work, the Meteorologica or ire pi /zerecooow. This
is the source of the famous extract made from Geminus by
Alexander Aphrodisiensis, and reproduced by Simplicius in
his commentary on the Physics of Aristotle, 2 on which Schia-
parelli relied in his attempt to show that it was Heraclides of
Pontus, not Aristarchus of Samos, who first put forward the
heliocentric hypothesis. The extract is on the distinction
between physical and astronomical inquiry as applied to the
heavens. It is the business of the physicist to consider the
substance of the heaven and stars, their force and quality,
their coming into being and decay, and he is in a position to
prove the facts about their size, shape, and arrangement;
astronomy, on the other hand, ignores the physical side,
proving the arrangement of the heavenly bodies by considera-
tions based on the view that the heaven is a real /coV/zoy, and,
when it tells us of the shapes, sizes and distances of the earth,
sun and moon, of eclipses and conjunctions, and of the quality
and extent of the movements of the heavenly bodies, it is
connected with the mathematical investigation of quantity,
size, form, or shape, and uses arithmetic and geometry to
prove its conclusions. Astronomy deals, not with causes, but
with facts ; hence it often proceeds by hypotheses, stating
certain expedients by which the phenomena may be saved.
For example, why do the sun, the moon and the planets
appear to move irregularly ? To explain the observed facts
we may assume, for instance, that the orbits are eccentric
circles or that the stars describe epicycles on a carrying
circle ; and then we have to go farther and examine other
ways in which it is possible for the phenomena to be brought
about. ' Hence we actually find a certain person [Heraclides
1 Proclus on Eucl. I, pp. 112. 22-113. 3, p. 251. 3-11.
2 Simpl. in Phys., pp. 291-2, ed. Diels.
232 SUCCESSORS OF THE GREAT GEOMETERS
of Pontus] coming forward and saying that, even on the
assumption that the earth moves in a certain ivay, while
the sun is in a certain way at rest, the apparent irregularity
with reference to the sun may be saved! Philological con-
siderations as well as the other notices which we possess
about Heraclides make it practically certain that ' Heraclides
of Pontus ' is an interpolation and that Geminus said tl?
simply, ' a certain person ', without any name, though he
doubtless meant Aristarchus of Samos. 1
Simplicius says that Alexander quoted this extract from
the epitome of the egijyrjo-is by Geminus. As the original
work was apparently made the subject of an abridgement, we
gather that it must have been of considerable scope. It is
a question whether egrjyrjcris means ' commentary ' or ' ex-
position ' ; I am inclined to think that the latter interpretation
is the correct one, and that Geminus reproduced Posidonius's
work in its entirety with elucidations and comments ; this
seems to me to be suggested by the words added by Simplicius
immediately after the extract ' this is the account given by
Geminus, or Posidonius in Geminus, of the difference between
physics and astronomy', which seems to imply that Geminus
in our passage reproduced Posidonius textually.
' Introduction to the Phaenomena ' attributed to Geminus.
There remains the treatise, purporting to be by Geminus,
which has come down to us under the title Teyiivov elaaycoyr)
els ra <f>ouv6neva. 2 What, if any, is the relation of this work
to the exposition of Posidonius's Meteorologica or the epitome
of it just mentioned? One view is that the original Isagoge
of Geminus and the e^rjyrjo-Ls of Posidonius were one and the
same work, though the Isagoge as we have it is not by
Geminus, but„by an unknown compiler. The objections to
this are, first, that it does not contain the extract given by
Simplicius, which would have come in usefully at the begin-
ning of an Introduction to Astrononi}^ nor the other extract
given by Alexander from Geminus and relating to the rainbow
which seems likewise to have come from the egijyrjo-Ls 3 ;
1 Cf. Aristarchus of Samos, pp. 275-83.
2 Edited by Manitius (Teubner, 1898).
3 Alex. Aphr. on Aristotle's Meteorologica, iii. 4, 9 (Ideler. ii, p. 128;
p. 152. 10, Hayduck).
GEMINUS 233
secondly, that it does not anywhere mention the name of
Posidonius (not, perhaps, an insuperable objection) ; and,
thirdly, that there are views expressed in it which are not
those held by Posidonius but contrary to them. Again, the
writer knows how to give a sound judgement as between
divergent views, writes in good style on the whole, and can
hardly have been the mere compiler of extracts from Posi-
donius which the view in question assumes him to be. It
seems in any case safer to assume that the Isagoge and the
egrjyrjo-is were separate works. At the same time, the Isagoge,
as we have it, contains errors which we cannot attribute to
Geminus. The choice, therefore, seems to lie between two
alternatives : either the book is by Geminus in the main, but
has in the course of centuries suffered deterioration by inter-
polations, mistakes of copyists, and so on, or it is a compilation
of extracts from an original Isagoge by Geminus with foreign
and inferior elements introduced either by the compiler him-
self or by other prentice hands. The result is a tolerable ele-
mentary treatise suitable for teaching purposes and containing
the most important doctrines of Greek astronomy represented
from the standpoint of Hipparchus. Chapter 1 treats of the
zodiac, the solar year, the irregularity of the sun's motion,
which is explained by the eccentric position of the sun's orbit
relatively to the zodiac, the order and the periods of revolution
of the planets and the moon. In § 23 we are told that all
the fixed stars do not lie on one spherical surface, but some
are farther away than others — a doctrine due to the Stoics.
Chapter 2, again, treats of the twelve signs of the zodiac,
chapter 3 of the constellations, chapter 4 of the axis of
the universe and the poles, chapter 5 of the circles on the
sphere (the equator and the parallel circles, arctic, summer-
tropical, winter- tropical, antarctic, the colure-circles, the zodiac
or ecliptic, the horizon, the meridian, and the Milky Way),
chapter 6 of Day and Night, their relative lengths in different
latitudes, their lengthening and shortening, chapter 7 of
the times which the twelve signs take to rise. Chapter 8
is a clear, interesting and valuable chapter on the calendar,
the length of months and years and the various cycles, the
octaeteris, the 16-years and 160-years cycles, the 19-years
cycle of Euctemon (and Meton), and the cycle of Callippus
234 SUCCESSORS OF THE GREAT GEOMETERS
(76 years). Chapter 9 deals with the moon's phases, chapters
10,11 with eclipses of the sun and moon, chapter 1 2 with the
problem of accounting for the motions of the sun, moon and
planets, chapter 13 with Risings and Settings and the various
technical terms connected therewith, chapter 14 with the
circles described by the fixed stars, chapters 15 and 16 with
mathematical and physical geography, the zones, &c. (Geminus
follows Eratosthenes's evaluation of the circumference of the
earth, not that of Posidonius). Chapter 17, on weather indica-
tions, denies the popular theory that changes of atmospheric
conditions depend on the rising and setting of certain stars,
and states that the predictions of weather (e7TLar)/j.a(riai.) in
calendars (irapaiTrjy fiara) are only derived from experience
and observation, and have no scientific value. Chapter 18 is
on the k^Xiyjios, the shortest period which contains an integral
number of synodic months, of days, and of anomalistic revolu-
tions of the moon ; this period is three times the Chaldaean
period of 223 lunations used for predicting eclipses. The end
of the chapter deals with the maximum, mean, and minimum
daily motion of the moon. The chapter as a whole does not
correspond to the rest of the book ; it deals with more difficult
matters, and is thought by Manitius to be a fragment only of
a discussion to which the compiler did not feel himself equal.
At the end of the work is a calendar (Parapegma) giving the
number of days taken by the sun to traverse each sign of
the zodiac, the risings and settings of various stars and the
weather indications noted by various astronomers, Democritus,
Eudoxus, Dositheus, Euctemon, Meton, Callippus ; this calendar
is unconnected with the rest of the book and the contents
are in several respects inconsistent with it, especially the
division of the year into quarters which follows^ Callippus
rather than Hipparchus. Hence it has been, since Boeckh's
time, generally considered not to be the work of Geminus.
Tittel, however, suggests that it is not impossible that Geminus
may have reproduced an older Parapegma of Callippus.
XVI
SOME HANDBOOKS
The description of the handbook on the elements of
astronomy entitled the Introduction to the Phaenomena and
attributed to Geminus might properly have been reserved
for this chapter. It was. however, convenient to deal with
Geminus in close connexion with Posidonius ; for Geminus
wrote an exposition of Posidonius's Meteorologica related to the
original work in such a way that Simplicius, in quoting a long
passage from an epitome of this work, could attribute the
passage to either Geminus or ' Posidonius in Geminus ' ; and it
is evident that, in other subjects too, Geminus drew from, and
was influenced by, Posidonius.
The small work De motu circulari corporum caelestium by
Cleomedes (KXeofirjSovs kvkXlktj Oecopfa) in two Books is the
production of a much less competent person, but is much more
largely based on Posidonius. This is proved by several refer-
ences to Posidonius by name, but it is specially true of the
very long first chapter of Book II (nearly half of the Book)
which seems for the most part to be copied bodily from
Posidonius, in accordance with the author's remark at the
end of Book I that, in giving the refutation of the Epicurean
assertion that the sun is just as large as it looks, namely one
foot in diameter, he will give so much as suffices for such an
introduction of the particular arguments used by 'certain
authors who have written whole treatises on this one topic
(i. e. the size of the sun), among whom is Posidonius '. The
interest of the book then lies mainly in what is quoted from
Posidonius ; its mathematical interest is almost ail.
The date of Cleomedes is not certainly ascertained, but, as
he mentions no author later than Posidonius, it is permissible
to suppose, with Hultsch, that he wrote about the middle of
236 SOME HANDBOOKS
the first century B. 0. As he seems to know nothing of the
works of Ptolemy, he can hardly, in any case, have lived
later than the beginning of the second century A. D.
Book I begins with a chapter the object of which is to
prove that the universe, which has the shape of a sphere,
is limited and surrounded by void extending without limit in
all directions, and to refute objections to this view. Then
follow chapters on the five parallel circles in the heaven and
the zones, habitable and uninhabitable (chap. 2) ; on the
motion of the fixed stars and the independent (irpoaiptTLKai)
movements of the planets including the sun and moon
(chap. 3); on the zodiac and the effect of the sun's motion in
it (chap. 4) ; on the inclination of the axis of the universe and
its effects on the lengths of days and nights at different places
(chap. 5); on the inequality in the rate of increase in the
lengths of the days and nights according to the time of year,
the different lengths of the seasons due to the motion of the
sun in an eccentric circle, the difference between a day-and-
night and an exact revolution of the universe owing to the
separate motion of the sun (chap. 6) ; on the habitable regions
of the globe including Britain and the ' island of Thule ', said
to have been visited by Pytheas, where, when the sun is in
Cancer and visible, the day is a month long ; and so on (chap. 7).
Chap. 8 purports to prove that the universe is a sphere by
proving first that the earth is a sphere, and then that the air
about it, and the ether about that, must necessarily make up
larger spheres. The earth is proved to be a sphere by the
method of exclusion ; it is assumed that the only possibilities
are that it is (a) flat and plane, or (b) hollow and deep, or
(c) square, or (d) pyramidal, or (e) spherical, and, the first four
hypotheses being successively disposed of, only the fifth
remains. Chap. 9 maintains that the earth is in the centre of
the universe ; chap. 10, on the size of the earth, contains the
interesting reproduction of the details of the measurements of
the earth by Posidonius and Eratosthenes respectively which
have been given above in their proper places (p. 220, pp. 1 06-7) ;
chap. 1 1 argues that the earth is in the relation of a point to,
i. e. is negligible in size in comparison with, the universe or
even the sun's circle, but not the moon's circle (cf. p. 3 above).
Book II, chap. 1, is evidently the 'piece de resistance, con-
CLEOMEDES 237
sisting of an elaborate refutation of Epicurus and his followers,
who held that the sun is just as large as it looks, and further
asserted (according to Cleomedes) that the stars are lit up as
they rise and extinguished as they set. The chapter seems to
be almost wholly taken from Posidonius ; it ends with some
pages of merely vulgar abuse, comparing Epicurus with Ther-
sites, with more of the same sort. The value of the chapter
lies in certain historical traditions mentioned in it, and in the
account of Posidonius's speculation as to the size and distance
of the sun, which does, as a matter of fact, give results much
nearer the truth than those obtained by Aristarchus, Hippar-
chus, and Ptolemy. Cleomedes observes (1) that by means of
water-clocks it is found that the apparent diameter of the sun
is 1/ 750th of the sun's circle, and that this method of
measuring it is said to have been first invented by the
Egyptians; (2) that Hipparchus is said to have found that
the sun is 1,050 times the size of the earth, though, as regards
this, we have the better authority of Adrastus (in Theon of
Smyrna) and of Chalcidius, according to whom Hipparchus
made the sun nearly 1,880 times the size of the earth (both
figures refer of course to the solid content). We have already
described Posidonius's method of arriving at the size and
distance of the sun (pp. 220-1). After he has given this, Cleo-
medes, apparently deserting his guide, adds a calculation of
his own relating to the sizes and distances of the moon and
the sun which shows how little he was capable of any scien-
tific inquiry. 1 Chap. 2 purports to prove that the sun is
1 He says (pp. 146. 17-148. 27) that in an eclipse the breadth of the
earth's shadow is stated to be two moon-breadths ; hence, he says, it
seems credible (tti6uv6u) that the earth is twice the size of the moon (this
practically assumes that the breadth of the earth's shadow is equal to
the diameter of the "earth, or that the cone of the earth's shadow is
a cylinder!). Since then the circumference of the earth, according to
Eratosthenes, is 250,000 stades, and its diameter therefore ' more than
80,000 ' (he evidently takes ir = 3), the diameter of the moon will be
40,000 stades. Now, the moon's circle being 750 times the moon's
diameter, the radius of the moon's circle, i.e. the distance of the moon
from the earth, will be |th of this (i.e. n = 3) or 125 moon-diameters;
therefore the moon's distance is 5,000,000 stades (which is much too
great). Again, since the moon traverses its orbit 13 times to the sun's
once, he assumes that the sun's orbit is 13 times as large as the moon's,
and consequently that the diameter of the sun is 13 times that of the
moon, or 520,000 stades and its distance 13 times 5,000,000 or 65,000,000
stades !
238 SOME HANDBOOKS
larger than the earth ; and the remaining chapters deal with
the size of the moon and the stars (chap. 3), the illumination
of the moon by the sun (chap. 4), the phases of the moon and
its conjunctions with the sun (chap. 5), the eclipses of the
moon (chap. 6), the maximum deviation in latitude of the five
planets (given as 5° for Venus, 4° for Mercury, 2|° for Mars
and Jupiter, 1° for Saturn), the maximum elongations of
Mercury and Venus from the sun (20° and 50° respectively),
and the synodic periods of the planets (Mercury 116 days,
Venus 584 days, Mars 780 days, Jupiter 398 days, Saturn
378 days) (chap. 7).
There is only one other item of sufficient interest to be
mentioned here. In Book II, chap. 6, Cleomedes mentions
that there were stories of extraordinary eclipses which ' the
more ancient of the mathematicians had vainly tried to ex-
plain '-£ the supposed 'paradoxical' case was that in which,
while the sun seems to be still above the horizon, the eclipsed
moon rises in the east. The passage has been cited above
(vol. i, pp. 6-7), where I have also shown that Cleomedes him-
self gives the true explanation of the phenomenon, namely
that it is due to atmospheric refraction.
The first and second centuries of the Christian era saw
a continuation of the work of writing manuals or introduc-
tions to the different mathematical subjects. About A. D. 100
came Nicomachus, who wrote an Introductio n to Arithmetic
and an Introduction to Harmony ; if we may judge by a
remark of his own, 1 he would appear to have written an intro-
duction to geometry also. The Arithmetical Introduction has
been sufficiently described above (vol. i, pp. 97-112).
There is yet another handbook which needs to be mentioned
separately, although we have had occasion to quote from it
several times already. This is the book by Theon OF Smyrna
which goes by the title Expositio rerum mathematicarum ad
legendum Platonem utilium. There are two main divisions
of this work, contained in two Venice manuscripts respec-
tively. The first was edited by Bullialdus (Paris, 1644), the
second by T. H. Martin (Paris, 1849); the whole has been
1 Nicom. Arith. ii. 6. 1.
THEON OF SMYRNA 239
edited by E. Hiller (Teubner, 1878) and finally, with a French
translation, by J. Dupuis (Paris, 1892).
Theon's date is approximately fixed by two considerations.
He is clearly the person whom Theon of Alexandria called
'the old Theon', top' iraXaibv Qecova} and there is no reason
to doubt that he is the ' Theon the mathematician ' (6 fiaOr)-
fianKos) who is credited by Ptolemy with four observations
of the planets Mercury and Venus made in a.d. 127, 129, 130
and 132. 2 The latest writers whom Theon himself mentions
are Thrasyllus, who lived in the reign of Tiberius, and
Adrastus the Peripatetic, who belongs to the middle of the
second century a.d. Theon's work itself is a curious medley,
valuable, not intrinsically, but for the numerous historical
notices which it contains. The title, which claims that the
book contains things useful for the study of Plato, must not
be taken too seriously. It was no doubt an elementary
introduction or vade-mecum for students of philosophy, but
there is little in it which has special reference to the mathe-
matical questions raised in Plato. The connexion consists
mostly in the long proem quoting the views of Plato on the
paramount importance of mathematics in the training of
the philosopher, and the mutual relation of the five different
branches, arithmetic, geometry, stereometry, astronomy and
music. The want of care shown by Theon in the quotations
from particular dialogues of Plato prepares us for the patch-
work character of the whole book.
In the first chapter he promises to give the mathematical
theorems most necessary for the student of Plato to know,
in arithmetic, music, and geometry, with its application to
stereometry and astronomy. 3 But the promise is by no means
kept as regards geometry and stereometry : indeed, in a
later passage Theon seems to excuse himself from including
theoretical geometry in his plan, on the ground that all those
who are likely to read his work or the writings of Plato may
be assumed to have gone through an elementary course of
theoretical geometry. 4 But he writes at length on figured
1 Theon of Alexandria, Comm. on Ptolemy s Syntaxis, Basel edition,
pp. 390, 395, 396.
2 Ptolemy, Syntaxis, ix. 9, x. 1, 2.
3 Theon of Smyrna, ed. Hiller, p. 1. 10-17.
4 lb., p. 16. 17-20.
240 SOME HANDBOOKS
numbers, plane and solid, which are of course analogous to
the corresponding geometrical figures, and he may have con-
sidered that he was in this way sufficiently fulfilling his
promise with regard to geometry and stereometry. Certain
geometrical definitions, of point, line, straight line, the three
dimensions, rectilinear plane and solid figures, especially
parallelograms and parallelepiped al figures including cubes,
plinthides (square bricks) and SoKiSes (beams), and scalene
figures with sides unequal every way (= ^(o/ilo-kol in the
classification of solid numbers), are dragged in later (chaps.
53-5 of the section on music) 1 in the middle of the discussion
of proportions and means ; if this passage is not an inter-
polation, it confirms the supposition that Theon included in
his work only this limited amount of geometry and stereo-
metry.
Section I is on Arithmetic in the same sense as Nicomachus's
Introduction. At the beginning Theon observes that arith-
metic will be followed by music. Of music in its three
aspects, music in instruments (ev opyavois), music in numbers,
i.e. musical intervals expressed in numbers or pure theoretical
music, and the music or harmony in the universe, the first
kind (instrumental music) is not exactly essential, but the other
two must be discussed immediately after arithmetic. 2 The con-
tents of the arithmetical section have been sufficiently indicated
in the chapter on Pythagorean arithmetic (vol. i, pp. 112-13) ;
it deals with the classification of numbers, odd, even, and
their subdivisions, prime numbers, composite numbers with
equal or unequal factors, plane numbers subdivided into
square, oblong, triangular and polygonal numbers, with their
respective ' gnomons ' and their properties as the sum of
successive terms of arithmetical progressions beginning with
1 as the first term, circular and spherical numbers, solid num-
bers with three factors, pyramidal numbers and truncated
pyramidal numbers, perfect numbers with their correlatives,
the over-perfect and the deficient; this is practically what
we find in Nicomachus. But the special value of Theon's
exposition lies in the fact that it contains an account of the
famous ' side- ' and ' diameter- ' numbers of the Pythagoreans. 3
1 Theon of Smyrna, ed. Hiller, pp. 111-13. 2 lb., pp. 16. 24-17. 11.
3 lb., pp. 42. 10-45. 9. Cf. vol. i, pp. 91-3.
THEON OF SMYRNA 241
In the Section on Music Theon says he will first speak of
the two kinds of music, the audible or instrumental, and the
intelligible or theoretical subsisting in numbers, after which
he promises to deal lastly with ratio as predicable of mathe-
matical entities in general and the ratio constituting the
harmony in the universe, ' not scrupling to set out once again
the things discovered by our predecessors, just as we have
given the things handed down in former times by the Pytha-
goreans, with a view to making them better known, without
ourselves claiming to have discovered any of them '. 1 Then
follows a discussion of audible music, the intervals which
give harmonies, &c, including substantial quotations from
Thrasyllus and Adrastus, and references to views of Aris-
toxenus, Hippasus, Archytas, Eudoxus and Plato. With
chap. 17 (p. 72) begins the account of the 'harmony in
numbers', which turns into a general discussion of ratios,
proportions and means, with more quotations from Plato,
Eratosthenes and Thrasyllus, followed by Thrasyllus's divisio
canonis, chaps. 35, 36 (pp. 87-93). After a promise to apply
the latter division to the sphere of the universe, Theon
purports to return to the subject of proportion and means.
This, however, does not occur till chap. 50 (p. 106), the
intervening chapters being taken up with a discussion of
the SeKois and rerpa/cTt/? (with eleven applications of the
latter) and the mystic or curious properties of the numbers
from 2 to 10; here we have a part of the theologumena of
arithmetic. The discussion of proportions and the different
kinds of means after Eratosthenes and Adrastus is again
interrupted by the insertion of the geometrical definitions
already referred to (chaps. 53-5, pp. 111-13), after which
Theon resumes the question of means for ' more precise '
treatment.
The Section on Astronomy begins on p. 120 of Killer's
edition. Here again Theon is mainly dependent upon
Adrastus, from whom he makes long quotations. Thus, on
the sphericity of the earth, he says that for the neces-
sary conspectus of the arguments it will be sufficient to
refer to the grounds stated summarily by Adrastus. In
explaining (p. 124) that the unevennesses in the surface of
1 Theon of Smyrna, ed. Hiller, pp. 46. 20-47. 14.
1523.2 R
242 SOME HANDBOOKS
the earth, represented e.g. by mountains, are negligible in
comparison with the size of the whole, he quotes Eratosthenes
and Dicaearchus as claiming to have discovered that the
perpendicular height of the highest mountain above the normal
level of the land is no more than 1 stades ; and to obtain the
diameter of the earth he uses Eratosthenes's figure of approxi-
mately 252,000 stades for" the circumference of the earth,
which, with the Archimedean value of - 2 T 2 - for 7r, gives a
diameter of about 80,182 stades. The principal astronomical
circles in the heaven are next described (chaps. 5-12, pp.
129-35) ; then (chap. 12) the assumed maximum deviations in
latitude are given, that of the sun being put at 1°, that of the
moon and Venus at 12°, and those of the planets Mercury,
Mars, Jupiter and Saturn at 8°, 5°, 5° and 3° respectively; the
obliquity of the ecliptic is given as the side of a regular polygon
of 15 sides described in a circle, i.e. as 24° (chap. 23, p. 151).
Next the order of the orbits of the sun, moon and planets is ex-
plained (the system is of course geocentric) ; we are told (p. 138)
that ' some of the Pythagoreans ' made the order (reckoning-
outwards from the earth) to be moon, Mercury, Venus, sun,
Mars, Jupiter, Saturn, whereas (p. 142) Eratosthenes put the
sun next to the moon, and the mathematicians, agreeing with
Eratosthenes in this, differed only in the order in which they
placed Venus and Mercury after the sun, some putting Mercury
next and some Venus (p. 143). The order adopted by ' some
of the Pythagoreans ' is the Chaldaean order, which was not
followed by any Greek before Diogenes of Babylon (second
century B.C.); 'some of the Pythagoreans' are therefore the
later Pythagoreans (of whom Nicomachus was one) ; the other
order, moon, sun, Venus, Mercury, Mars, Jupiter, Saturn, was
that of Plato and the early Pythagoreans. In chap. 15
(p. 138 sq.) Theon quotes verses of Alexander 'the Aetolian'
(not really the ' Aetolian ', but Alexander of Ephesus, a con-
temporary of Cicero, or possibly Alexander of Miletus, as
Chalcidius calls him) assigning to each of the planets (includ-
ing the earth, though stationary) with the sun and moon and
the sphere of" the fixed stars one note, the intervals between
the notes being so arranged as to bring the nine into an
octave, whereas with Eratosthenes and Plato the earth was
excluded, and the eight notes of the octachord were assigned
THEON OF SMYRNA 243
to the seven heavenly bodies and the sphere of the fixed stars.
The whole of this passage (chaps. 15 to 16, pp. 138-47) is no
doubt intended as the promised account of the ' harmony in
the universe ', although at the very end of the work Theon
implies that this has still to be explained on the basis of
Thrasyllus's exposition combined with what he has already
given himself.
The next chapters deal with the forward movements, the
stationary points, and the retrogradations, as they respectively
appear to us, of the five planets, and the ' saving of the pheno-
mena ' by the alternative hypotheses of eccentric circles and
epicycles (chaps. 17-30, pp. 147-78). These hypotheses are
explained, and the identity of the motion produced by the
two is shown by Adrastus in the case of the sun (chaps. 26, 27,
pp. 166-72). The proof is introduced with the interesting
remark that ' Hipparchus says it is worthy of investigation
by mathematicians why, on two hypotheses so different from
one another, that of eccentric circles and that of concentric
circles with epicycles, the same results appear to follow '. It
is not to be supposed that the proof of the identity could be
other than easy to a mathematician like Hipparchus ; the
remark perhaps merely suggests that the two hypotheses were
discovered quite independently, and it was not till later that
the effect was discovered to be the same, when of course the
fact would seem to be curious and a mathematical proof would
immediately be sought. Another passage (p. 188) says that
Hipparchus preferred the hypothesis of the epicycle, as being
his own. If this means that Hipparchus claimed to have
discovered the epicycle-hypothesis, it must be a misapprehen-
sion ; for Apollonius already understood the theory of epi-
cycles in all its generality. According to Theon, the epicycle-
hypothesis is more ' according to nature ' ; but it was presum-
ably preferred because it was applicable to all the planets,
whereas the eccentric-hypothesis, when originally suggested,
applied only to the three superior planets ; in order to make
it apply to the inferior planets it is necessary to suppose the
circle described by the centre of the eccentric to be greater
than the eccentric circle itself, which extension of the hypo-
thesis, though known to Hipparchus, does not seem to have
occurred to Apollonius.
R 2
244 SOME HANDBOOKS
We next have (chap. 31, p. 178) an allusion to the systems
of Eudoxus, Callippus and Aristotle, and a description
(p. 180 sq.) of a system in which the 'carrying' spheres
(called ' hollow ') have between them ' solid spheres which by
their own motion will roll (dveXigovcri) the carrying spheres in
the opposite direction, being in contact with them \ These
' solid ' spheres (which carry the planet fixed at a point on
their surface) act in practically the same way as epicycles.
In connexion with this description Theon (i.e. Adrastus)
speaks (chap. 33, pp. 186-7) of two alternative hypotheses in
which, by comparison with Chalcidius, 1 we recognize (after
eliminating epicycles erroneously imported into both systems)
the hypotheses of Plato and Heraclides respectively. It is
this passage which enables us to conclude for certain that
Heraclides made Venus and Mercury revolve in circles about
the sun, like satellites, while the sun in its turn revolves in
a circle about the earth as centre. Theon (p. 187) gives the
maximum arcs separating Mercury and Venus respectively
from the sun as 20° and 50°, these figures being the same as
those given by Cleomedes.
The last chapters (chaps. 37-40), quoted from Adrastus, deal
with conjunctions, transits, occultations and eclipses. The
book concludes with a considerable extract from Dercy Hides,
a Platonist with Pythagorean leanings, who wrote (before the
time of Tiberius and perhaps even before Varro) a book on
Plato's philosophy. It is here (p. 198. 14) that we have the
passage so often quoted from Eudemus :
' Eudemus relates in his Astronomy that it was Oenopides
who first discovered the girdling of the zodiac and the revolu-
tion (or cycle) of the Great Year, that Thales was the first to
discover the eclipse of the sun and the fact that the sun's
period with respect to the solstices is not always the same,
that Anaximander discovered that the earth is (suspended) on
high and lies (substituting Keircu for the reading of the manu-
scripts, KLveirai, moves) about the centre of the universe, and
that Anaximenes said that the moon has its light from the
sun and (explained) how its eclipses come about' (Anaxi-
menes is here apparently a mistake for Anaxagoras).
1 Chalcidius, Comm. on Timaens, c. 110. Cf. Aristarchus of Samos,
pp. 256-8.
XVII
TRIGONOMETRY: HIPPARCHUS, MENELAUS,
PTOLEMY
We have seen that Sphaeric, the geometry of the sphere,
was very early studied, because it was required so soon as
astronomy became mathematical ; with the Pythagoreans the
word Sphaeric, applied to one of the subjects of the quadrivium,
actually meant astronomy. The subject was so far advanced
before Euclid's time that there was in existence a regular
textbook containing the principal propositions about great
and small circles on the sphere, from which both Autolycus
and Euclid quoted the propositions as generally known.
These propositions, with others of purely astronomical in-
terest, were collected afterwards in a work entitled Sphaerica,
in three Books, by Theodosius.
Suidas has a notice, s. v. QeoSocnos, which evidently con-
fuses the author of the Sphaerica with another Theodosius,
a Sceptic philosopher, since it calls him ' Theodosius, a philoso-
pher ', and attributes to him, besides the mathematical works,
' Sceptic chapters ' and a commentary on the chapters of
Theudas. Now the commentator on Theudas must have
belonged, at the earliest, to the second half of the second
century A.D., whereas our Theodosius was earlier than Mene-
laus {fi. about A. D. 100), who quotes him by name. The next
notice by Suidas is of yet another Theodosius, a poet, who
came from Tripolis. Hence it was at one time supposed that
our Theodosius was of Tripolis. But Vitruvius x mentions a
Theodosius who invented a sundial ? for any climate ' ; and
Strabo, in speaking of certain Bithynians distinguished in
their particular sciences, refers to ' Hipparchus, Theodosius
and his sons, mathematicians ' 2 . We conclude that our Theo-
1 Be architectura ix. 9. 2 Strabo, xii. 4, 9, p. 566.
246 TRIGONOMETRY
dosius was of Bithynia and not later in date than Vitruvius
(say 20 B.C.); but the order in which Strabo gives the
names makes it not unlikely that he was contemporary with
Hipparchus, while the character of his Sphaerica suggests a
date even earlier rather than later.
Works by Theodosius.
Two other works of Theodosius besides the Sphaerica,
namely On habitations and On Days and Nights, seem to
have been included in the 'Little Astronomy' [jxiKpos d&Tpo-
vofjiovfj.ei'os, sc. tottos). These two treatises need not detain us
long. They are extant in Greek (in the great MS. Vaticanus
Graecus 204 and others), but the Greek text has not appar-
ently yet been published. In the first, On habitations, in 12
propositions, Theodosius explains the different phenomena due
to the daily rotation of the earth, and the particular portions
of the whole system which are visible to inhabitants of the
different zones. In the second, On Days and Nights, contain-
ing 13 and 19 propositions in the two Books respectively,
Theodosius considers the arc of the ecliptic described by the
sun each day, with a view to determining the conditions to be
satisfied in order that the solstice may occur in the meridian
at a given place, and in order that the day and the night may
really be equal at the equinoxes; he shows also that the
variations in the day and night must recur exactly after
a certain time, if the length of the solar year is commen-
surable with that of the day, while on the contrary assump-
tion they will not recur so exactly.
In addition to the works bearing on astronomy, Theodosius
is said l to have written a commentary, now lost, on the e<p6Siov
or Method of Archimedes (see above, pp. 27-34).
Contents of the Sphaerica.
We come now to the Sphaerica, which deserves a short
description from the point of view of this chapter. A text-
book on the geometry of the sphere was wanted as a supple-
ment to the Elements of Euclid. In the Elements themselves
1 Suidas, loc. cit.
THEODOSIUS'S SPHAERIGA 247
(Books XII and XIII) Euclid included no general properties
of the sphere except the theorem proved in XII. 16-18, that
the volumes of two spheres are in the triplicate ratio of their
diameters ; apart from this, the sphere is only introduced in
the propositions about the regular solids, where it is proved
that they are severally inscribable in a sphere, and it was doubt-
less with a view to his proofs of this property in each case that
he gave a new definition of a sphere as the figure described by
the revolution of a semicircle about its diameter, instead of
the more usual definition (after the manner of the definition
of a circle) as the locus of all points (in space instead of in
a plane) which are equidistant from a fixed point (the centre).
No doubt the exclusion of the geometry of the sphere from
the Elements was due to the fact that it was regarded as
belonging to astronomy rather than pure geometry.
Theodosius defines the sphere as ' a solid figure contained
by one surface such that all the straight lines falling upon it
from one point among those lying within the figure are equal
to one another ', which is exactly Euclid's definition of a circle
with ' solid ' inserted before ' figure ' and ' surface ' substituted
for ' line '. The early part of the work is then generally
developed on the lines of Euclid's Book III on the circle.
Any plane section of a sphere is a circle (Prop. 1). The
straight line from the centre of the sphere to the centre of
a circular section is perpendicular to the plane of that section
(1, Por. 2 ; cf. 7, 23); thus a plane section serves for finding
the centre of the sphere just as a chord does for finding that
of a circle (Prop. 2). The propositions about tangent planes
(3-5) and the relation between the sizes of circular sections
and their distances from the centre (5, 6) correspond to
Euclid III. 16-19 and 15; as the small circle corresponds to
any chord, the great circle (' greatest circle ' in Greek) corre-
sponds to the diameter. The poles of a circular section
correspond to the extremities of the diameter bisecting
a chord of a circle at right angles (Props. 8-10). Great
circles bisecting one another (Props. 11-12) correspond to
chords which bisect one another (diameters), and great circles
bisecting small circles at right angles and passing through
their poles (Props. 13-15) correspond to diameters bisecting
chords at right angles. The distance of any point of a great
248 TRIGONOMETRY
circle from its pole is equal to the side of a square inscribed
in the great circle and conversely (Props. 16, 17). Next come
certain problems : To find a straight line equal to the diameter
of any circular section or of the sphere itself (Props. 18, 19) ;
to draw the great circle through any two given points on
the surface (Prop. 20) ; to find the pole of any given circu-
lar section (Prop. 21). Prop. 22 applies Eucl. III. 3 to the
sphere.
Book II begins with a definition of circles on a sphere
which touch one another ; this happens ' when the common
section of the planes (of the circles) touches both circles '.
Another series of propositions follows, corresponding again
to propositions in Eucl., Book III, for the circle. Parallel
circular sections have the same poles, and conversely (Props.
1, 2). Props. 3-5 relate to circles on the sphere touching
one another and therefore having their poles on a great
circle which also passes through the point of contact (cf.
Eucl. III. 11, [12] about circles touching one another). If
a great circle touches a small circle, it also touches another
small circle equal and parallel to it (Props. 6, 7), and if a
great circle be obliquely inclined to another circular section,
it touches each of two equal circles parallel to that section
(Prop. 8). If two circles on a sphere cut one another, the
great circle drawn through their poles bisects the intercepted
segments of the circles (Prop. 9). If there are any number of
parallel circles on a sphere, and any number of great circles
drawn through their poles, the arcs of the parallel circles
intercepted between any two of the great circles are similar,
and the arcs of the great circles intercepted between any two
of the parallel circles are equal (Prop. 10).
The last proposition forms a sort of transition to the portion
of the treatise (II. 11-23 and Book III) which contains pro-
positions of purely astronomical interest, though expressed as
propositions in pure geometry without any specific reference
to the various circles in the heavenly sphere. The proposi-
tions are long and complicated, and it would neither be easy
nor worth while to attempt an enumeration. They deal with
circles or parts of circles (arcs intercepted on one circle by
series of other circles and the like). We have no difficulty in
recognizing particular circles which come into many proposi-
THEODOSIUS'S SPHAERIGA 249
tions. A particular small circle is the circle which is the
limit of the stars which do not set, as seen by an observer at
a particular place on the earth's surface ; the pole of this
circle is the pole in the heaven. A great circle which touches
this circle and is obliquely inclined to the ' parallel circles ' is the
circle of the horizon ; the parallel circles of course represent
the apparent motion of the fixed stars in the diurnal rotation,
and have the pole of the heaven as pole. A second great
circle obliquely inclined to the parallel circles is of course the
circle of the zodiac or ecliptic. The greatest of the ' parallel
circles ' is naturally the equator. All that need be said of the
various propositions (except two which will be mentioned
separately) is that the sort of result proved is like that of
Props. 12 and 13 of Euclid's Phaenomena to the effect that in
the half of the zodiac circle beginning with Cancer (or Capri-
cornus) equal arcs set (or rise) in unequal times ; those which
are nearer the tropic circle take a longer time, those further
from it a shorter; those which take the shortest time are
those adjacent to the equinoctial points ; those which are equi-
distant from the equator rise and set in equal times. In like
manner Theodosius (III. 8) in effect takes equal and con-
tiguous arcs of the ecliptic all on one side of the equator,
draws through their extremities great circles touching the
circumpolar ' parallel ' circle, and proves that the correspond-
ing arcs of the equator intercepted between the latter great
circles are unequal and that, of the said arcs, that correspond-
ing to the arc of the ecliptic which is nearer the tropic circle
is the greater. The successive great circles touching the
circumpolar circle are of course successive positions of the
horizon as the earth revolves about its axis, that is to say,
the same length of arc on the ecliptic takes a longer or shorter
time to rise according as it is nearer to or farther from the
tropic, in other words, farther from or nearer to the equinoctial
points.
It is, however, obvious that investigations of this kind,
which only prove that certain arcs are greater than others,
and do not give the actual numerical ratios between them, are
useless for any practical purpose such as that of telling the
hour of the night by the stars, which was one of the funda-
mental problems in Greek astronomy ; and in order to find
250
TRIGONOMETRY
the required numerical ratios a new method had to be invented,
namely trigonometry.
No actual trigonometry in Theodosius.
It is perhaps hardly correct to say that spherical triangles
are nowhere referred to in Theodosius, for in III. 3 the con-
gruence-theorem for spherical triangles corresponding to Eucl.
I. 4 is practically proved ; but there is nothing in the book
that can be called trigonometrical. The nearest approach is
in III. 11, 12, where ratios between certain straight lines are
compared with ratios between arcs. ACc (Prop. 11) is a great
circle through the poles A, A' ; CDc, CD are two other great
circles, both of which are at right angles to the plane of ACc,
but CDc is perpendicular to A A\ while CD is inclined to it at
an acute angle. Let any other great circle AB'BA' through
A A' cut CD in any point B between C and D, and CD in B'.
Let the ' parallel ' circle EB'e be drawn through B\ .and let
Cc r be the diameter of the ' parallel ' circle touching the great
circle CD. Let L, K be the centres of the ' parallel ' circles,
and let R, p be the radii of the ' parallel ' circles CDc, Cc f
respectively. It is required to prove that
2R:2p> (arc CB) : (arc CB r ).
Let CO, Ee meet in N, and join NB'.
Then B'N, being the intersection of two planes perpendicu-
lar to the plane of ACCA f , is perpendicular to that plane and
therefore to both Ee and CO.
THEODOSIUS'S SPHAERICA 251
Now, the triangle NLO being right-angled at L, NO > NL.
Measure NT along NO equal to NL, and join TB'.
Then in the triangles B'NT, B f NL two sides B'N, NT are
equal to two sides B'N, NL, and the included angles (both
being right) are equal ; therefore the triangles are equal in all
respects, and lNLB' = LNTB'.
Now 2R:2p = OC':C'K
= ON:NL
= ON:NT
[= tun NTB'itim NOB']
> A NTS': I NOB'
> INLB':/_N0B'
> IC0B:IN0B'
> (arc.BC):(arc.B'<7').
If a', b\ c' are the sides of the spherical triangle AB'C, this
result is equivalent (since the angle COB subtended by the arc
OB is equal to A) to
1 : sin b' = tan A : tan a'
> a : a',
where a = BG, the side opposite A in the triangle ABC.
The proof is based on the fact (proved in Euclid's Optics
and assumed as known by Aristarchus of Samos and Archi-
medes) that, if a, f3 are angles such that \ tt > oc > /3,
tan a/tan f3 > ol/$.
While, therefore, Theodosius proves the equivalent of the
formula, applicable in the solution of a spherical triangle
right-angled at C, that tana = sin 6 tan J., he is unable, for
want of trigonometry, to find the actual value of a/ a', and
can only find a limit for it. He is exactly in the same position
as Aristarchus, who can only approximate to the values of the
trigonometrical ratios which he needs, e.g. sin 1°, cos 1°, sin 3°,
by bringing them within upper and lower limits with the aid
of the inequalities
tan oc <x sin a
tan /? /3 sin /3 '
where -| n > oc > /3.
252 TRIGONOMETRY
We may contrast with this proposition of Theodosius the
corresponding proposition in Menelaus's Sphaerica (III. 15)
dealing with the more general case in which C", instead of
being the tropical point on the ecliptic, is, like B' , any point
between the tropical point and D. If R, p have the same
meaning as above and r x , r 2 are the radii of the parallel circles
through B' and the new C\ Menelaus proves that
sina Rp
sin a' r Y r 2 '
which, of course, with the aid of Tables, gives the means
of finding the actual values of a or a! when the other elements
are given.
The proposition III. 12 of Theodosius proves a result similar
to that of III. 11 for the case where the great circles AB'B,
AC'C, instead of being great circles through the poles, are
great circles touching ' the circle of the always-visible stars ',
i.e. different positions of the horizon, and the points C", B f are
any points on the arc of the oblique circle between the tropical
and the equinoctial points ; in this case, with the same notation,
4E : 2 p > (arc BG) : (arc B'C).
It is evident that Theodosius was simply a laborious com-
piler, and that there was practically nothing original in his
work. It has been proved, by means of propositions quoted
verbatim or assumed as known by Autolycus in his Moving
Sphere and by Euclid in his Phaenomena, that the following
propositions in Theodosius are pre-Euclidean, I. 1, 6 a, 7, 8, 11,
12, 13, 15, 20 ; II. 1, 2, 3, 5, 8, 9, 10 a, 13, 15, 17, 18, 19, 20, 22;
III. lb, 2, 3, 7, 8. those shown in thick type being quoted
word for word.
The beginnings of trigonometry.
But this is not all. In Menelaus's Spliaerica, III. 15, there
is a reference to the proposition (III. 11) of Theodosius proved
above, and in Gherard of Cremona's translation from the
Arabic, as well as in Halley's translation from the Hebrew
of Jacob b. Machir, there is an addition to the effect that this
proposition was used by Apollonius in a book the title of
which is given in the two translations in the alternative
BEGINNINGS OF TRIGONOMETRY 253
forms ' liber aggregativus ' and ' liber cle principiis universa-
libus'. Each of these expressions may well mean the work
of Apollonius which Marinus refers to as the ' General
Treatise ' (fj kccOoXov wpayfjiaTeia). There is no apparent
reason to doubt that the remark in question was really
contained in Menelaus's original work ; and, even if it is an
Arabian interpolation, it is not likely to have been made
without some definite authority. If then Apollonius was the
discoverer of the proposition, the fact affords some ground for
thinking that the beginnings of trigonometry go as far back,
at least, as Apollonius. Tannery 1 indeed suggested that not
only Apollonius but Archimedes before him may have com-
piled a ' table of chords ', or at least shown the way to such
a compilation, Archimedes in the work of which we possess
only a fragment in the Measurement of a Circle^ and Apollonius
in the cdkvtoklov, where he gave an approximation to the value
of tt closer than that obtained by Archimedes; Tannery
compares the Indian Table of Sines in the Surya-Siddhdnta,
where the angles go by 24ths of a right angle (l/24th = 3° 45',
2/24ths=7° 30', &c), as possibly showing Greek influence.
This is, however, in the region of conjecture ; the first person
to make systematic use of trigonometry is, so far as we know,
Hipparchus.
Hipparchus, the greatest astronomer of antiquity, was
born at Nicaea in Bithynia. The period of his activity is
indicated by references in Ptolemy to observations made by
him the limits of which are from 161 B.C. to 126 B.C. Ptolemy
further says that from Hipparchus's time to the beginning of
the reign of Antoninus Pius (a.d. 138) was 265 years. 2 The
best and most important observations made by Hipparchus
were made at Rhodes, though an observation of the vernal
equinox at Alexandria on March 24, 146 B.C., recorded by him
may have been his own. His main contributions to theoretical
and practical astronomy can here only be indicated in the
briefest manner.
1 Tannery, Recherches sur Vhist. de Vastronomie ancienne, p. 64.
2 Ptolemy, Syntaxis, vii. 2 (vol. ii, p. 15).
254 TRIGONOMETRY
The work of Hipparchus.
Discovery of precession.
1. The greatest is perhaps his discovery of the precession
of the equinoxes. Hipparchus found that the bright star
Spica was, at the time of his observation of it, 6° distant
from the autumnal equinoctial point, whereas he deduced from
observations recorded by Timocharis that .Timocharis had
made the distance 8°. Consequently the motion had amounted
to 2° in the period between Timocharis's observations, made in
283 or 295 B.C., and 129/8 B.C., a period, that is, of 154 or
166 years; this gives about 46-8" or 43-4" a year, as compared
with the true value of 50-3757".
Calculation of mean lunar month.
2. The same discovery is presupposed in his work On the
length of the Year, in which, by comparing an observation
of the summer solstice by Aristarchus in 281/0 B.C. with his
own in 136/5 B.C., he found that after 145 years (the interval
between the two dates) the summer solstice occurred half
a day-and-night earlier than it should on the assumption of
exactly 365J days to the year; hence he concluded that the
tropical year contained about ^§o^ n °f a day-and-night less
than 3 65 \ days. This agrees very nearly with Censorinus's
statement that Hipparchus's cycle was 304 years, four times
the 76 years of Callippus, but with 111,035 days in it
instead of 111,036 ( = 27,759x4). Counting in the 304 years
12x304 + 112 (intercalary) months, or 3,760 months in all,
Hipparchus made the mean lunar month 29 days 12 hrs.
44 min. 2-| sec, which is less than a second out in comparison
with the present accepted figure of 29-53059 days!
3. Hipparchus attempted a new determination of the sun's
motion by means of exact equinoctial and solstitial obser-
vations; he reckoned the eccentricity of the sun's course
and fixed the apogee at the point 5° 30' of Gemini. More
remarkable still was his investigation of the moon's
course. He determined the eccentricity and the inclination
of the orbit to the ecliptic, and by means of records of
observations of eclipses determined the moon's period with
extraordinary accuracy (as remarked above). We now learn
HIPPARCHUS 255
that the lengths of the mean synodic, the sidereal, the
anomalistic and the draconitic month obtained by Hipparchus
agree exactly with Babylonian cuneiform tables of date not
later than Hipparchus, and it is clear that Hipparchus was
in full possession of all the results established by Babylonian
astronomy.
Improved estimates of sizes and distances of sun
and moon.
4. Hipparchus improved on Aristarchus's calculations of the
sizes and distances of the sun and moon, determining the
apparent diameters more exactly and noting the changes in
them ; he made the mean distance of the sun 1,245 D, the mean
distance of the moon 33|D, the diameters of the sun and
moon 1 2 J D and J D respectively, where D is the mean
diameter of the earth.
Epicycles and eccentrics.
5. Hipparchus, in investigating the motions of the sun, moon
and planets, proceeded on the alternative hypotheses of epi-
cycles and eccentrics ; he did not invent these hypotheses,
which were already fully understood and discussed by
Apollonius. While the motions of the sun and moon could
with difficulty be accounted for by the simple epicycle and
eccentric hypotheses, Hipparchus found that for the planets it
was necessary to combine the two, i.e. to superadd epicycles to
motion in eccentric circles.
Catalogue of stars.
6. He compiled a catalogue of fixed stars including 850 or
more such stars; apparently he was the first to state their
positions in terms of coordinates in relation to the ecliptic
(latitude and longitude), and his table distinguished the
apparent sizes of the stars. His work was continued by
Ptolemy, who produced a catalogue of 1,022 stars which,
owing to an error in his solar tables affecting all his longi-
tudes, has by many erroneously been supposed to be a mere
reproduction of Hipparchus's catalogue. That Ptolemy took
many observations himself seems certain. 1
1 See two papers by Dr. J. L. E. Dreyer in] the Monthly Notices of the
Royal Astronomical Society, 1917, pp. 528-39, and 1918.. pp. 343-9.
256 TRIGONOMETRY
Improved Instruments.
7. He made great improvements in the instruments used for
observations. Among those which he used were an improved
dioptra, a ' meridian-instrument ' designed for observations in
the meridian only, and a universal instrument {acrTpoXdPov
opyavov) for more general use. He also made a globe on
which he showed the positions of the fixed stars as determined
by him ; it appears that he showed a larger number of stars
on his globe than in his catalogue.
Geography.
In geography Hipparchus wrote a criticism of Eratosthenes,
in great part unfair. He checked Eratosthenes's data by
means of a sort of triangulation ; he insisted on the necessity
of applying astronomy to geography, of fixing the position of
places by latitude and longitude, and of determining longitudes
by observations of lunar eclipses.
Outside the domain of astronomy and geography, Hipparchus
wrote a book On things borne doivn by their weight from
which Simplicius (on Aristotle's De caelo, p. 264 sq.) quotes
two propositions. It is possible, however, that even in this
work Hipparchus may have applied his doctrine to the case of
the heavenly bodies.
In pure mathematics he is said to have considered a problem
in permutations and combinations, the problem of finding the
number of different possible combinations of 10 axioms or
assumptions, which he made to be 103,049 (v. I. 101,049)
or 310,952 according as the axioms were affirmed or denied 1 :
it seems impossible to make anything of these figures. When
the Fihrist attributes to him works ' On the art of algebra,
known by the title of the Rules ' and ' On the division of num-
bers ', we have no confirmation : Suter suspects some confusion,
in view of the fact that the article immediately following in
the Fihrist is on Diophantus, who also ' wrote on the art of
algebra \
1 Plutarch, Quaest. Conviv, viii. 9. 3, 732 f, De Stoicorum repugn. 29.
1047 d.
HIPPARCHUS 257
First systematic use of Trigonometry.
We come now to what is the most important from the
point of view of this work, Hipparchus's share in the develop-
ment of trigonometry. Even if he did not invent it,
Hipparchus is the first person of whose systematic use of
trigonometry we have documentary evidence. (1) Theon
of Alexandria says on the Syntaxis of Ptolemy, a propos of
Ptolemy's Table of Chords in a circle (equivalent to sines),
that Hipparchus, too, wrote a treatise in twelve books on
straight lines (i.e. chords) in a circle, while another in six
books was written by Menelaus. 1 In the Syntaxis I. 10
Ptolemy gives the necessary explanations as to the notation
used in his Table. The circumference of the circle is divided
into 360 parts or degrees; the diameter is also divided into
120 parts, and one of such parts is the unit of length in terms
of which the length of each chord is expressed ; each part,
whether of the circumference or diameter, is divided into 60
parts, each of these again into 60, and so on, according to the
system of sexagesimal fractions. Ptolemy then sets out the
minimum number of propositions in plane geometry upon
which the calculation of the chords in the Table is based (8ia
rijs e/c tcov ypafifioou fjieOoSiKrjs ccvtcov avo-Tcicrecos). The pro-
positions are famous, and it cannot be doubted that Hippar-
chus used a set of propositions of the same kind, though his
exposition probably ran to much greater length. As Ptolemy
definitely set himself to give the necessary propositions in the
shortest form possible, it will be better to give them under
Ptolemy rather than here. (2) Pappus, in speaking of Euclid's
propositions about the inequality of the times which equal arcs
of the zodiac take to rise, observes that ' Hipparchus in his book
On the rising of the twelve signs of the zodiac shows by means
of numerical calc%dations (6Y dpiOfioou) that equal arcs of the
semicircle beginning with Cancer which set in times having
a certain relation to one another do not everywhere show the
same relation between the times in which they rise ', 2 and so
on. We have seen that Euclid, Autolycus, and even Theo-
dosius could only prove that the said times are greater or less
1 Theon, Comm. on Syntaxis, p. 110, ed. Halma.
2 Pappus, vi, p. 600. 9-13.
1523.2 S
258 TRIGONOMETRY
in relation to one another ; they could not Calculate the actual
times. As Hipparchus proved corresponding propositions by
means of numbers, we can only conclude that he used proposi-
tions in spherical trigonometry, calculating arcs from others
which are given, by means of tables. (3) In the only work
of his which survives, the Commentary on the Phaenomena
of Eudoxus and Aratus (an early work anterior to the
discovery of the precession of the equinoxes), Hipparchus
states that (presumably in the latitude of Rhodes) a star which
lies 27^° north of the equator describes above the horizon an
arc containing 3 minutes less than 15/24ths of the whole
circle 1 ; then, after some more inferences, he says, ' For each
of the aforesaid facts is proved by means of lines (8ia rS>v
ypafificov) in the general treatises on these matters compiled
by me '. In other places 2 of the Commentary he alludes to
a work On simultaneous risings (ra irepl tcov o-vvavaroXcov),
and in II. 4. 2 he says he will state summarily, about each of
the fixed stars, along with what sign of the zodiac it rises and
sets and from which degree to which degree of each sign it
rises or sets in the regions about Greece or wherever the
longest day is 14^ equinoctial hours, adding that he has given
special proofs in another work designed so that it is possible
in practically every place in the inhabited earth to follow
the differences between the concurrent risings and settings. 3
Where Hipparchus speaks of proofs ' by means of lines ', he
does not mean a merely graphical method, by construction
only, but theoretical determination by geometry, followed by
calculation, just as Ptolemy uses the expression e/c tS>v ypa/i-
/jlcov of his calculation of chords and the expressions cr(paipiKal
SeigeLs and ypafifiLKal Seigeis of the fundamental proposition
in spherical trigonometry (Menelaus's theorem applied to the
sphere) and its various applications to particular cases. It
is significant that in the Syntaxis VIII. 5, where Ptolemy
applies the proposition to the very problem of finding the
times of concurrent rising, culmination and setting of the
fixed stars, he says that the times can be obtained ' by lines
only ' (8ia ixovoav tcov ypafifi(£>v). A Hence we may be certain
that, in the other books of his own to which Hipparchus refers
1 Ed. Manitius, pp. 148-50. 2 lb., pp. 128. 5, 148. 20.
3 lb., pp. 182. 19-184. 5. 4 Syntaxis, vol. ii, p. 193.
HIPPARCHUS 259
in his Commentary, he used the formulae of spherical trigono-
metry to get his results. In the particular case where it is
required to find the time in which a star of 27-§° northern
declination describes, in the latitude of Rhodes, the portion of
its arc above the horizon, Hipparchus must have used the
equivalent of the formula in the solution of a right-angled
spherical triangle, tan b = cos A tan c, where C is the right
angle. Whether, like Ptolemy, Hipparchus obtained the
formulae, such as this one, which he used from different
applications of the one general theorem (Menelaus's theorem)
it is not possible to say. There was of course no difficulty
in calculating the tangent or other trigonometrical function
of an angle if only a table of sines was given ; for Hippar-
chus and Ptolemy were both aware of the fact expressed by
sin 2 a + cos 2 a = 1 or, as they would have written it,
(crd. 2a) 2 + {crd. (180°-2a)} 2 = 4r 2 ,
where (crd. 2 a) means the chord subtending an arc 2 a, and r
is the radius, of the circle of reference.
Table of Chords.
We have no details of Hipparchus's Table of Chords suffi-
cient to enable us to compare it with Ptolemy's, which goes
by half-degrees, beginning with angles of |°, 1°, l-§°, and so
on. But Heron 1 in his Metrica says that 'it is proved in the
books about chords in a circle ' that, if « 9 and a n are the sides
of a regular enneagon (9 -sided figure) and hendecagon (1 1 -sided
figure) inscribed in a circle of diameter d, then (1) a 9 = ^d,
(2) a u = £gd very nearly, which means that sin 20° was
taken as equal to 0-3333 ... (Ptolemy's table makes it
Rn( 20 "** fin + fin?)' S0 ^ a ^ ^ e ^ rs ^ a PP roxmia ti° n is §), and
sin T X T . 180° or sin 16° 21' 49" was made equal to 0-28 (this cor-
responds to the chord subtending an angle of 32° 43' 38",nearly
half-way between 32J° and 33°, and the mean between the two
1 /lt> 54 55 \
chords subtending the latter angles gives — ( + — H | as
the required sine, while eV ( 16 A) = Iff, which only differs
1 Heron, Metrica, i. 22, 24, pp. 58. 19 and 62. 17.
s2
260 TRIGONOMETRY
by ¥ Jo from ■§§§ or -^ T , Heron's figure). There is little doubt
that it is to Hipparchus's work that Heron refers, though the
author is not mentioned.
While for our knowledge of Hipparchus's trigonometry we
have to rely for the most part upon what we can infer from
Ptolemy, we fortunately possess an original source of infor-
mation about Greek trigonometry in its highest development
in the Sphaerica of Menelaus.
The date of Menelaus of Alexandria is roughly indi-
cated by the fact that Ptolemy quotes an observation of
his made in the first year of Trajan's reign (a.d. 98). He
was therefore- a contemporary of Plutarch, who in fact
represents him as being present at the dialogue De facie in
orbe lunae, where (chap. 17) Lucius apologizes to Menelaus 'the
mathematician ' for questioning the fundamental proposition
in optics that the angles of incidence and reflection are equal.
He wrote a variety of treatises other than the Sphaerica.
We have seen that Theon mentions his work on Chords in a
Circle in six Books. Pappus says that he wrote a treatise
(Trpayixareia) on the setting (or perhaps only rising) of
different arcs of the zodiac. 1 Proclus quotes an alternative
proof by him of Eucl. I. 25, which is direct instead of by
reductio ad absurdum, 2, and he would seem to have avoided
the latter kind of proof throughout. Again, Pappus, speaking
of the many complicated curves ' discovered by Demetrius of
Alexandria (in his " Linear considerations ") and by Philon
of Tyana as the result of interweaving plectoids and other
surfaces of all kinds ', says that one curve in particular was
investigated by Menelaus and called by him ' paradoxical '
(irapd8o£os) 3 ; the nature of this curve can only be conjectured
(see below).
But Arabian tradition refers to other works by Menelaus,
(l) Elements of Geometry, edited by Thabit b. Qurra, in three
Books, (2) a Book on triangles, and (3) a work the title of
which is translated by Wenrich de cognitione quantitatis
discretae corporum permixtomm. Light is thrown on this
last title by one al-Chazini who (about A.D. 1121) wrote a
1 Pappus, vi, pp. 600-2.
2 Proclus on Eucl. I, pp. 345. 14-346. 11.
3 Pappus, iv, p. 270. 25.
MENELAUS OF ALEXANDRIA 261
treatise about the hydrostatic balance, i.e. about the deter-
mination of the specific gravity of homogeneous or mixed
bodies, in the course of which he mentions Archimedes and
Menelaus (among others) as authorities on the subject; hence
the treatise (3) must have been a book on hydrostatics dis-
cussing such problems as that of the crown solved by Archi-
medes. The alternative proof of Eucl. I. 25 quoted by
Proclus might have come either from the Elements of Geometry
or the Book on triangles. With regard to the geometry, the
' liber trium fratrum ' (written by three sons of Musa b. Shakir
in the ninth century) says that it contained a solution of the
duplication of the cube, which is none other than that of
Archytas. The solution of Archytas having employed the
intersection of a tore and a cylinder (with a cone as well),
there would, on the assumption that Menelaus reproduced the
solution, be a certain appropriateness in the suggestion of
Tannery 1 that the curve which Menelaus called the napd8o£os
ypa/jL/xi] was in reality the curve of double curvature, known
by the name of Viviani, which is the intersection of a sphere
with a cylinder touching it internally and having for its
diameter the radius of the sphere. This curve is a particular
case of Eudoxus's hipiDopede, and it has the property that the
portion left outside the curve of the surface of the hemisphere
on which it lies is equal to the square on the diameter of the
sphere ; the fact of the said area being squareable would
justify the application of the word napdSogos to the curve,
and the quadrature itself would not probably be beyond the
powers of the Greek mathematicians, as witness Pappus's
determination of the area cut off between a complete turn of
a certain spiral on a sphere and the great circle touching it at
the origin. 2
The Sphaerica of Menelaus.
This treatise in three Books is fortunately preserved in
the Arabic, and although the extant versions differ con-
siderably in form, the substance is beyond doubt genuine ;
the original translator was apparently Ishaq b. Hunain
(died A. D. 910). There have been two editions, (1) a Latin
1 Tannery, Memoires scientifiqites^ ii, p. 17. 2 Pappus, iv, pp. 264-8.
262 TRIGONOMETRY
translation by Maurolycus (Messina, 1558) and (2) Halley's
edition (Oxford, 1758). The former is unserviceable because
Maurolycus' s manuscript was very imperfect, and, besides
trying to correct and restore the propositions, he added
several of his own. Halley seems to have made a free
translation of the Hebrew version of the work by Jacob b.
Machir (about 1273), although he consulted Arabic manuscripts
to some extent, following them, e.g., in dividing the work into
three Books instead of two. But an earlier version direct
from the Arabic is available in manuscripts of the thirteenth
to fifteenth centuries at Paris and elsewhere ; this version is
without doubt that made by the famous translator Gherard
of Cremona (1114-87). With the help of Halley's edition,
Gherard's translation, and a Leyden manuscript (930) of
the redaction of the work by Abu-Nasr-Mansur made in
A.D. 1007-8, Bjornbo has succeeded in presenting an adequate
reproduction of the contents of the Sphaerica. 1
Book I.
In this Book for the first time we have the conception and
definition of a spherical triangle. Menelaus does not trouble
to give the usual definitions of points and circles related to
the sphere, e.g. pole, great circle, small circle, but begins with
that of a spherical triangle as ' the area included by arcs of
great circles on the surface of a sphere ', subject to the restric-
tion (Def. 2) that each of the sides or legs of the triangle is an
arc less than a semicircle. The angles of the triangle are the
angles contained by the arcs of great circles on the sphere
(Def. 3), and one such angle is equal to or greater than another
according as the planes containing the arcs forming the first
angle are inclined at the same angle as, or a greater angle
than, the planes of the arcs forming the other (Defs. 4, 5).
The angle is a right angle if the planes of the arcs are at right
angles (Def. 6). Pappus tells us that Menelaus in his Sphaerica
calls the figure in question (the spherical triangle) a ' three-
side ' (rpnrXefpo^) 2 ; the word triangle (Tpiyatvov) was of course
i
1 Bjornbo, Studien uber Menelaos' Spharik (Abhandlungen zur Gesch. d.
math. Wissenschaften,Heft xiv. 1902).
2 Pappus, vi, p. 476. 16.
MENELAUS'S SPHAERICA 263
already appropriated for the plane triangle. We should gather
from this, as well as from the restriction of the definitions to
the spherical triangle and its parts, that the discussion of the
spherical triangle as such was probably new ; and if the pre-
face in the Arabic version addressed to a prince and beginning
with the words, ' prince ! I have discovered an excellent
method of proof . . . ' is genuine, we have confirmatory evidence
in the writer's own claim.
Menelaus's object, so far as Book I is concerned, seems to
have been to give the main propositions about spherical
triangles corresponding to Euclid's propositions about plane
triangles. At the same time he does not restrict himself to
Euclid's methods of proof even where they could be adapted
to the case of the sphere ; he avoids the form of proof by
reductio ad absurdum, but, subject to this, he prefers the
easiest proofs. In some respects his treatment is more com-
plete than Euclid's treatment of the analogous plane cases.
In the congruence-theorems, for example, we have I. 4 a
corresponding to Eucl. I. 4, I. 4b to Eucl. I. 8, I. 14, 16 to
Eucl. I. 26 a, b; but Menelaus includes (I. 13) what we know
as the ' ambiguous case ', which is enunciated on the lines of
Eucl. VI. 7. I. 12 is a particular case of I. 16. Menelaus
includes also the further case which has no analogue in plane
triangles, that in which the three angles of one triangle are
severally equal to the three angles of the other (1.17). He
makes, moreover, no distinction between the congruent and
the symmetrical, regarding both as covered by congruent. 1. 1
is a problem, to construct a spherical angle equal to a given
spherical angle, introduced only as a lemma because required
in later propositions. I. 2, 3 are the propositions about
isosceles triangles corresponding to Eucl. I. 5, 6 ; Eucl. 1. 18, 19
(greater side opposite greater angle and vice versa) have their
analogues in I. 7, 9, and Eucl. I. 24, 25 (two sides respectively
equal and included angle, or third side, in one triangle greater
than included angle, or third side, in the other) in I. 8. I. 5
(two sides of a triangle together greater than the third) corre-
sponds to Eucl. I. 20. There is yet a further group of proposi-
tions comparing parts of spherical triangles, I. 6, 18, 19, where
I. 6 (corresponding to Eucl. I. 21) is deduced from I. 5, just as
the first part of Eucl. I. 21 is deduced from Eucl. I. 20.
264 TRIGONOMETRY
Eucl. I. 16, 32 are not true of spherical triangles, and
Menelaus has therefore the corresponding but different pro-
positions. I. 10 proves that, with the usual notation a, b, c,
A, B, 0, for the sides and opposite angles of a spherical
triangle, the exterior angle at C, or 180° — G, < = or >A
according as c + a> = or < 180°, and vice versa. The proof
of this and the next proposition shall be given as specimens.
In the triangle ABC suppose that c + a > = or < 180° ; let
D be the pole opposite to A.
Then, according as c + a > = or < 180°, BC > = or < BD
(since AD = 180°),
and therefore ID > = or < I BCD (= 180°-C), [I. 9]
i.e. (since ID = LA) 180°- G < = or > A.
Menelaus takes the converse for granted.
As a consequence of this, I. 1 1 proves that A + B + C> 180°.
Take the same triangle ABG, with the pole D opposite
to A, and from B draw the great circle BE such that
Z.DBE = IBDE.
Then GE+EB = CD < 180°, so that, by the preceding
proposition, the exterior angle AGB to the triangle BGE is
greater than LGBE y
i.e. C>ACBE.
Add A dv D (= IEBD) to the unequals;
therefore G + A > L GBD,
whence A + B + C > IGBD + B or 180°.
After two lemmas I. 21, 22 we have some propositions intro-
ducing M, N, P the middle points of a, 6, c respectively. I. 23
proves, e.g., that the arc MN of a great circle >-|c, and I. 20
that AM < = or > \a according as A > = or < (B + C). The
last group of propositions, 26-35, relate to the figure formed
MENELAUS'S SPHAERICA 265
by the triangle ABC with great circles drawn through B to
meet AC (between A and G) in D, E respectively, and the
case where D and E coincide, and they prove different results
arising from different relations between a and c (a > c), com-
bined with the equality of A D and EG (or DC), of the angles
ABD and EBG (or DBG), or of a + c and BD + BE (or 2BD)
respectively, according as a + c< = or >180°.
Book II has practically no interest for us. The object of it
is to establish certain propositions, of astronomical interest
only, which are nothing more than generalizations or exten-
sions of propositions in Theodosius's Sphaerica, Book III.
Thus Theodosius III. 5, 6, 9 are included in Menelaus II. 10,
Theodosius III. 7-8 in Menelaus II. 12, while Menelaus II. 11
is an extension of Theodosius III. 13. The proofs are quite
different from those of Theodosius, which are generally very
long-winded.
Book III. Trigonometry.
It will have been noticed that, while Book I of Menelaus
gives the geometry of the spherical triangle, neither Book I
nor Book II contains any trigonometry. This is reserved for
Book III. As I shall throughout express the various results
obtained in terms of the trigonometrical ratios, sine, cosine,
tangent, it is necessary to explain once for all that the Greeks
did not use this terminology, but, instead of sines, they used
the chords subtended by arcs of a
circle. In the accompanying figure
let the arc iDof a circle subtend an
angle a at the centre 0. Draw AM
perpendicular to OD, and produce it
to meet the circle again in A' . Then
sin a = AM/AO, and AM is \AA'
or half the chord subtended by an
angle 2 a at the centre, which may
shortly be denoted by J(crd. 2 a).
Since Ptolemy expresses the chords as so many 120th parts of
the diameter of the circle, while AM / AO — AA'/2A0, it
follows that sin a and J(crd. 2 a) are equivalent. Cos a is
of course sin (90° — a) and is therefore equivalent to % crd.
(180°-2a).
266
TRIGONOMETRY
(a) ' Menelaus s theorem ' for the sphere.
The first proposition of Book III is the famous ' Menelaus's
theorem ' with reference to a spherical triangle and any trans-
versal (great circle) cutting the sides of a triangle, produced
if necessary. Menelaus does not, however, use a spherical
triangle in his enunciation, but enunciates the proposition in
terms of intersecting great circles. ' Between two arcs ADB,
AEG of great circles are two other arcs of great circles DFG
and BFE which intersect them and also intersect each other
in F. All the arcs are less than a semicircle. It is required
to prove that
sin CE sin CF sin DB ,
sin EA " sin FD sin BA
It appears that Menelaus gave three or four cases, sufficient
to prove the theorem completely. The proof depends on two
simple propositions which Menelaus assumes without proof;
the proof of them is given by Ptolemy.
(1) In the figure on the last page, if OD be a radius cutting
a chord AB in C, then
AC:CB = sin AD: sin DB.
For draw A 31, BN perpendicular to OD. Then
AG:GB = AM:BN
= |(crd. 2.4D):i(crd. 2DB)
= sin AD: sin DB.
(2) If AB meet the radius OC produced in T, then
AT:BT = sin AC: sin BC.
MENELAUS'S SPIIAERICA 267
For, if AM, BN are perpendicular to OC, we have, as before,
AT:TB = AM:BN
= !(crd. 2^6'):i(crd. 2BC)
= sin.J_C':sini?G Y .
Now let the arcs of great circles ADB, A EC be cut by the
arcs of great circles DFC, BFE which themselves meet in F.
Let G be the centre of the sphere and join GB, GF, GE, AD.
Then the straight lines AD, GB, being in one plane, are
either parallel or not parallel. If they are not parallel, they
will meet either in the direction of D, B or of A, G.
Let AD, GB nieet in T.
Draw the straight lines ARC, DLC meeting GE, GF in K, L
respectively.
Then K, L, T must lie on a straight line, namely the straight
line which is the section of the planes determined by the arc
EFB and by the triangle AGD. 1
Thus we have two straight lines AC, AT cut by the two
straight lines CD, TK which themselves intersect in L.
Therefore, by Menelaus's proposition in plane geometry,
CK CL DT
KA~ LD'TA
1 So Ptolemy. In other words, since the straight lines GB, GE, GF,
which are in one plane, respectively intersect the* straight lines AD, AC,
CD which are also in one plane, the points of intersection T, K, L are in
both planes, and therefore lie on the straight line in which the planes
intersect.
268 TRIGONOMETRY
But, by the propositions proved above,
GK sin GE GL sin GF DT _ sin DB
KA ~ sin EA' LD ~~ sin FD* YA ~ sJn~BA '
therefore, by substitution, we have
sin CE __ sin (LP sin DB
sin EA " sin FD ' sin BA "
Menelaus apparently also gave the proof for the cases in
which J.Z), (ri? meet towards A, G, and in which AD, GB are
parallel respectively, and also proved that in like manner, in
the above figure,
sin GA sin CD sin FB
sin AE sin DF sin BE
(the triangle cut by the transversal being here CFE instead of
ADG). Ptolemy 1 gives the proof of the above case only, and
dismisses the last-mentioned result with a ' similarly '.
(/3) Deductions from Menelaus s Theorem.
III. 2 proves, by means of I. 14, 10 and III. 1, that, if ABC,
A'B'G' be two spherical triangles in which A — A', and G, G f
are either equal or supplementary, sin c/sin a = sin c'/sin a'
and conversely. The particular case in which G, (7 are right
angles gives what was afterwards known as the ' regula
quattuor quantitatum ' and was fundamental in Arabian
trigonometry. 2 A similar association attaches to the result of
III. 3, which is the so-called ' tangent ' or ' shadow-rule ' of the
Arabs/ If ABC, A f B'G' be triangles right-angled at A, A', and
G, G f are equal and both either > or < 90°, and if P, P f be
the poles of AG, A'C, then
sin AB _ sinA'B' sin BP
sin AG ~ sin A'G' ' sin B'P' '
Apply the triangles so that G' falls on C, C'B' on GB as GE,
and C A' on GA as GD ; then the result follows directly from
III. 1. Since sin BP — cos AB, and sin B'P' = cos A'B\ the
result becomes
sin GA tan AB
sin C'A' " ta^rZ 7 ^ 5
which is the ' tangent-rule ' of the Arabs. 3
1 Ptolemy, Syntax-is, i. 13, vol. i, p. 76.
2 See Braunmuhl, Gesch. der Trig, i, pp. 17, 47, 58-60, 127-9.
3 Cf. Braunmuhl, op. cit. i, pp. 17-18, 58, 67-9, &c.
MENELAUS'S SPHAERICA
269
It follows at once (Prop. 4) that, if AM, A'M' are great
circles drawn perpendicular to the bases BG, B'C of two
spherical triangles ABC, A'B'C in which B = B',C — G',
sin BM sin MC / . , , , , , tan AM \
— — ^i^r, — ~ — TF77T* I since both are equal to jttF' )'
sin B'M' smM'C'K * tan A'M'}
III. 5 proves that, if there are two spherical triangles ABC,
P P'
A'B'C right-angled at A, A' and such that C—C, while 6
and 6' are less than 90°,
sin (a + b) _ sin (a' + &')
sin (a — b) sin (a/ — b')
from which we may deduce 1 the formula
sin (a + b) 1 + cos 6 T
sin (a — b) ~" 1 — cos C
which is equivalent to tan b = tan a cos C.
(y) Anharmonic property of four great circles through
one point.
But more important than the above result is the fact that
the proof assumes as known the anhar-
monic property of four great circles
drawn from a point on a sphere in rela-
tion to any great circle intersecting them
all, viz. that, if ABCD, A'B'G'D' be two
transversals,
sin AD sin BC sinA'D' sin B'C
sin DG ' sin AB " sin B'C' ' sin A'B' '
* Braunmiihl, op. cit. i, p. 18; Bjornbo, p. 96.
270 TRIGONOMETRY
It follows that this proposition was known before Mene-
laus's time. It is most easily proved by means of ' Menelaus's
Theorem', III. 1, or alternatively it may be deduced for the
sphere from the corresponding proposition in plane geometry,
just as Menelaus's theorem is transferred by him from the
plane to the sphere in III. 1. We may therefore fairly con-
clude that both the anharmonic property and Menelaus's
theorem with reference to the sphere were already included
in some earlier text-book ; and, as Ptolemy, who built so much
upon Hipparchus, deduces many of the trigonometrical
formulae which he uses from the one theorem (III. 1) of
Menelaus, it seems probable enough that both theorems were
known to Hipparchus. The corresponding plane theorems
appear in Pappus among his lemmas to Euclid's Porisms, 1 and
there is therefore every probability that they were assumed
by Euclid as known.
(8) Projoositions analogous to Eucl. VI. S.
Two theorems following, III. 6, 8, have their analogy in
Eucl. VI. 3. In III. 6 the vertical angle i of a spherical
triangle is bisected by an arc of a great circle meeting BG in
D, and it is proved that sin BD/ sin DC = sin BA/ sin AC;
in III. 8 we have the vertical angle bisected both internally
and externally by arcs of great circles meeting BC in D and
E, and the proposition proves the harmonic property
sin BE sin BD
sin EC sin DC
III. 7 is to the effect that, if arcs of great circles be drawn
through B to meet the opposite side AC of a spherical triangle
in D, E so that lABD = L EBC, then
sin EA . sin A D _ sin 2 AB
sin DC . sin CE ~ sin 2 J30'
As this is analogous to plane propositions given by Pappus as
lemmas to different works included in the Treasury of
Analysis, it is clear that these works were familiar to
Menelaus.
1 Pappus, vii, pp. 870-2, 874.
MENELAUS'S SPHAERICA 271
III. 9 and III. 10 show, for a spherical triangle, that (1) the
great circles bisecting the three angles, (2) the great circles
through the angular points meeting the opposite sides at
right angles meet in a point.
The remaining propositions, III. 11-15, return to the same
sort of astronomical problem as those dealt with in Euclid's
Phaenomena, Theodosius's Sphaerica and Book II of Mene-
laus's own work. Props. 11-14 amount to theorems in
spherical trigonometry such as the following.
Given arcs a 1} a 2 , 0f 3 , a 4 , fi x , p 2 , /3 3 , /2 4 , such that
90° ^oc x > a 2 >a 3 >a 4 ,
90° >t3 1 >t3 2 >(3 3 >t3 4 ,
and also oc l >/3 1 , a 2 >/3 2 , a 3 >/3 3 , a 4 >/? 4 ,
(1) If sin a x : sin oc 2 : sin a 3 : sin cx 4 = sin /? x : sin /? 2 : sin /3 3 : sin/? 4 ,
then *=*>%=£'.
(2) If siD ("i + ft) _ shi(a 2 + ft 2 ) _ 8 in(« 3 + /8 3 )
sin (o^-ft) " sin (a 2 -0 2 ) "" sin(a 3 -/? 3 )
s in fa 4 + ft 4 )
' sin(a 4 -/3 4 )'
a i~ a 2 ^ ft-ft
then • ^ ? <
a 3-«4 P3-P4
If sin (o^-o^) < sin _(ft-£ 2 )
sin (a 3 — a 4 ) sin (£ 3 - /2 4 )
then *^ < ^ •
a 3-«4 ^3-^4
Again, given three series of three arcs such that
ol 1 >ol 2 >ol 33 @ 1 >P 2 >P B , 90°>y 1 >y 2 >y 3 ,
and sin (otj — y x ) : sin (a 2 — y 2 ) : sin (a 3 — y 3 )
= sin (ft - y T ) : sin (0 2 - y 2 ) : sin (0 3 - y 3 )
= siny 1 :siny 2 :siny 3
272 TRIGONOMETRY
(1) If * 1 >P 1 >2y 1 , a 2 >0 2 >2y 2 , a 3 >/? 3 >2y 3 ,
then ai ~ a2 >f 1_ ^ 2 ; and
a 2-a 3 ^2-/^3
(2) If P 1 <<x 1 <y 1 , P 2 <0( 2 < y 2 , 3 < a 3 < y 3 ,
then
a T — a 5
ol — a.
<
ft -ft
ft" ft
III. 15, the last proposition, is in four parts. The first part
is the proposition corresponding to Theodosius III. 11 above
alluded to. Let BA, BG be two quadrants of great circles
(in which we easily recognize the equator and the ecliptic),
P the pole of the former, PA 19 PA 3 quadrants of great circles
meeting the other quadrants in A 19 A z and C lf G z respectively.
Let R be the radius of the sphere, r, r lt r 3 the radii of the
' parallel circles ' (with pole P) through C, C, , C 3 respectively.
Then shall
sin AiA 3 __ Rr
sin G 1 C Z " r x r s
In the triangles PGG 3 , BA Z C% the angles at G, A 3 are right,
and the angles at C 3 equal ; therefore (III. 2)
sin PG sin BA
mmm o
sin PC Q sin BC\
MENELAUS'S SPHAERICA 273
But, by III. 1 applied to the triangle BC\A l cut by the
transversal PC A..,
sinJ.jJ.3 sin C^C 3 sin PA T
sin BA 3 sin BC 3 sin PC\
sin A, A- sin PA sinBA., sin PA sin PC
sin C 1 C 3 sin PC Y sin BC 3 sin PC 1 sin PC.
3
from above,
Br
7\?
1'3
Part 2 of the proposition proves that, if PC' 2 ^. 2 be drawn
such that sin 2 PC 2 = sin PA 2 . sin PC, or r 2 2 = Rr (where r 2 is
the radius of the parallel circle through (7 2 ), BC 2 —BA 2 is a
maximum, while Parts 3, 4 discuss the limits to the value of
the ratio between the arcs A^A 3 and Cfi^.
Nothing is known of the life of Claudius Ptolemy except
that he was of Alexandria, made observations between the
years a.d. 125 and 141 or perhaps 151, and therefore presum-
ably wrote his great work about the middle of the reign of
Antoninus Pius (a.d. 138-61). A tradition handed down by
the Byzantine scholar Theodorus Meliteniota (about 1361)
states that he was born, not at Alexandria, but at Ptolemais
17 'Epfieiov. Arabian traditions, going back probably to
Hunain b. Ishaq, say that he lived to the age of 78, and give
a number of personal details to which too much weight must
not be attached.
The MaOrj/iaTLKT) o-vvTagts (Arab. Almagest).
Ptolemy's great work, the definitive achievement of Greek
astronomy, bore the title MaOrj/xaTiKrjs Zwrdgem /3i/3\ia i-y,
the Mathematical Collection in thirteen Books. By the time
of the commentators who distinguished the lesser treatises on
astronomy forming an introduction to Ptolemy's work as
fXLKpb? dcrTpoyo/jiov/iepo? (7677-09), the 'Little Astronomy', the
book came to be called the ' Great Collection ', fieydXrj <tvv-
tcl£is. Later still the Arabs, combining the article Al with
1523.? T
274 TRIGONOMETRY
the superlative /xeyicrTos, made up a word Al-majisti, which
became Almagest ; and it has been known by this name ever
since. The complicated character of the system expounded
by Ptolemy is no doubt responsible for the fact that it
speedily became the subject of elaborate commentaries.
Commentaries on the Syntaxis.
Pappus 1 cites a passage from his own commentary on
Book I of the Mathematica, which evidently means Ptolemy's
work. Part of Pappus's commentary on Book V, as well as
his commentary on Book VI, are actually extant in the
original. Theon of Alexandria, who wrote a commentary on
the Syntaxis in eleven Books, incorporated as much as was
available of Pappus's commentary on Book V with full
acknowledgement, though not in Pappus's exact words. In
nis commentary on Book VI Theon made much more partial
quotations from Pappus ; indeed the greater part of the com-
mentary on this Book is Theon's own or taken from other
sources. Pappus's commentaries are called scholia, Theon's
virofiurj/xaTa. Passages in Pappus's commentary on Book V
allude to ' the scholia preceding this one ' (in the plural), and
in particular to the scholium on Book IV. It is therefore all
but certain that he wrote on all the Books from I to VI at
least. The text of the eleven Books of Theon's commentary
was published at Basel by Joachim Camerarius in 1538, but
it is rare and, owing to the way in which it is printed, with
insufficient punctuation marks, gaps in places, and any number
of misprints, almost unusable ; accordingly little attention has
so far been paid to it except as regards the first two Books,
which were included, in a more readable form and with a Latin
translation, by Halma in his edition of Ptolemy.
Translations and editions.
The Syntaxis was translated into Arabic, first (we are told)
by translators unnamed at the instance of Yahya b. Khalid b.
Barmak, then by al-Hajjaj, the translator of Euclid (about
786-835), and again by the famous translator Ishaq b. Hunain
(d. 910), whose translation, as improved by. Thabit b. Qurra
1 Pappus, vhi, p. 1106. 13.
PTOLEMY'S SYNTAXIS ' 275
(died 901), is extant in part, as well as the version by Nasirad-
din at-TusI (1201-74).
The first edition to be published was the Latin translation
made by Gherard of Cremona from the Arabic, which was
finished in 1175 but was not published till 1515, when it was
brought out, without the author's name, by Peter Liechten-
stein at Venice. A translation from the Greek had been made
about 1160 by an unknown writer for a certain Henricus
Aristippus, Archdeacon of Catania, who, having been sent by
William I, King of Sicily, on a mission to the Byzantine
Emperor Manuel I. Comnenus in 1158, brought back with
him a Greek manuscript of the Syntaxis as a present; this
translation, however, exists only in manuscripts in the Vatican
and at Florence. The first Latin translation from the Greek
to be published was that made by Georgius ' of Trebizond ' for
Pope Nicolas V in 1451 ; this was revised and published by
Lucas Gauricus at Venice in 1528. The editio princeps of the
Greek text was brought out by Grynaeus at Basel in 1538.
The next complete edition was that of Halma published
1813-16, which is now rare. All the more welcome, there-
fore, is the definitive Greek text of the astronomical works
of Ptolemy edited by Heiberg (1899-1907), to which is now
added, so far as the Syntaxis is concerned, a most valuable
supplement in the German translation (with notes) by Manitius
(Teubner, 1912-13).
Summary of Contents.
The Syntaxis is most valuable for the reason that it con-
tains very full particulars of observations and investigations
by Hipparchus, as well as of the earlier observations recorded
by him, e.g. that of a lunar eclipse in 721 B.C. Ptolemy
based himself very largely upon Hipparchus, e.g. in the
preparation of a Table of Chords (equivalent to sines), the
theory of eccentrics and epicycles, &c. ; and it is questionable
whether he himself contributed anything of great value except
a definite theory of the motion of the five planets, for which
Hipparchus had only collected material in the shape of obser-
vations made by his predecessors and himself. A very short
indication of the subjects of the different Books is all that can
T 2
276 TRIGONOMETRY
bo given here. Book I : Indispensable preliminaries to the
study of the Ptolemaic system, general explanations of
the different motions of the heavenly bodies in relation to
the earth as centre, propositions required for the preparation
of Tables of Chords, the Table itself, some propositions in
spherical geometry leading to trigonometrical calculations of
the relations of arcs of the equator, ecliptic, horizon and
meridian, a ' Table of Obliquity ', for calculating declinations
for each degree-point on the ecliptic, and finally a method of
finding the right ascensions for arcs of the ecliptic equal to
one-third of a sign or 10°. Book II: The same subject con-
tinued, i.e. problems on the sphere, with special reference to
the differences between various latitudes, the length of the
longest day at any degree of latitude, and the like. Book III :
On the length of the year and the motion of the sun on the
eccentric and epicycle hypotheses. Book IV : The length of the
months and the theory of the moon. Book V : The construc-
tion of the astrolabe, and the theory of the moon continued,
the diameters of the sun, the moon and the earth's shadow,
the distance of the sun and the dimensions of the sun, moon
and earth. Book VI : Conjunctions and oppositions of sun
and moon, solar and lunar eclipses and their periods. Books
VII and VIII are about the fixed stars and the precession of
the equinoxes, and Books IX-XIII are devoted to the move-
ments of the planets.
Trigonometry in Ptolemy.
What interests the historian of mathematics is the trigono-
metry in Ptolemy. It is evident that no part of the trigono-
metry, or of the matter preliminary to it, in Ptolemy was new.
What he did was to abstract from earlier treatises, and to
condense into the smallest possible space, the minimum of
propositions necessary to establish the methods and formulae
used. Thus at the beginning of the preliminaries to the
Table of Chords in Book I he says :
1 We will first show how we can establish a systematic and
speedy method of obtaining the lengths of the chords based on
the uniform use of the smallest possible number of proposi-
tions, so that we may not only have the lengths of the chords
PTOLEMY'S SYNTAXIS 277
set out correctly, but may be in possession of a ready proof of
our method of obtaining them based on geometrical con-
sideration^' 1
He explains that he will use the division (1) of the circle into
360 equal parts or degrees and (2) of the diameter into 120
equal parts, and will express fractions of these parts on the
sexagesimal system. Then come the geometrical propositions,
as follows.
(a) Lemma for finding sin 18° and sin 36°.
To find the side of a pentagon and decagon inscribed in
a circle or, in other words, the chords subtending arcs of 72°
and 36° respectively.
Let AB be the diameter of a circle, the centre, OC the
radius perpendicular to AB.
Bisect OB at D, join DC, and measure
DE along DA equal to DC. Join EC.
Then shall OE be the side of the in-
scribed regular decagon, and EC the side
of the inscribed regular pentagon.
For, since OB is bisected at D,
BE.E0 + 0D 2 = DE 2
= DC 2 =D0 2 + 0C 2 .
Therefore BE. E0 = OC 2 = OB 2 ,
and BE is divided in extreme and mean ratio.
But (Eucl. XIII. 9) the sides of the regular hexagon and the
regular decagon inscribed in a circle when placed in a straight
line with one another form a straight line divided in extreme
and mean ratio at the point of division.
Therefore, BO being the side of the hexagon, EO is the side
of the decagon.
Also (by Eucl. XIII. 10)
(side of pentagon) 2 = (side of hexagon) 2 + (side of decagon) 2
= CO 2 + OE 2 = EC 2 ;
therefore EC is the side of the regular pentagon inscribed
in the circle.
*
1 Ptolemy, Syntaxis, i. 10, pp. 31 2.
278 TRIGONOMETRY
The construction in fact easily leads to the results
EO = \a( a/5-1), EC = %a</(lO-2V5), ,
where a is the radius of the circle.
Ptolemy does not however use these radicals, but calculates
the lengths in terms of ' parts ' of the diameter thus.
DO = 30, and DO 2 = 900 ; OG = 60 and OG 2 = 3600 ;
therefore BE 2 = DC 2 = 4500,' and DE = 67" 4' 55" nearly ;
therefore side of decagon or (crd. 36°)=DE-D0 = 37P4' 55".
Again OE 2 = (37 P 4' 55") 2 = 1375 . 4' 15", and OC 2 = 3600 ;
therefore C# 2 = 4975 . 4' 15", and 0# = 70^ 32' 3" nearly,
i.e. side of pentagon or (crd. 72°) = 702 J 32' 3".
The method of extracting the square root is explained by
Tlieon in connexion with the first of these cases, \/4500 (see
above, vol. i, pp. 61-3).
The chords which are the sides of other regular inscribed
figures, the hexagon, the square and the equilateral triangle,
are next given, namely,
crd. 60° = 60^,
crd. 90° = \/(2 . 60 2 ) = 7(7200) = 84^ 51' 10",
crd. 120° = a/(3 . 60 2 ) = 7(10800) = 103^ 55' 23".
(f$) Equivalent of sin 2 + cos 2 = 1.
It is next observed that, if x be any arc,
(crd. a;) 2 + {drd. (180°— a)} 2 = (diam.) 2 = 120 2 ,
a formula which is of course equivalent to sin 2 6 + cos 2 = 1 .
We can therefore, from crd. 72°, derive crd. 108°, from
crd. 36°, crd. 144°, and so on.
(y) ' Ptolemy's theorem ', giving the equivalent of
sin (6 — <p) = sin cos (j> r-r cos sin 0.
The next step is to find a formula which will give us
crd. (ot — fi) when crd. a and crd. p are given. (This for
instance enables us to find crd. 12° from crd. 72° and crd. 60°.)
PTOLEMY'S SYNTAXIS
279
The proposition giving the required formula depends upon
a lemma, which is the famous ' Ptolemy's theorem '.
Given a quadrilateral A BCD inscribed in a circle, the
diagonals being AG, BD, to prove that
AC.BD = AB.DC+AD.BC.
The proof is well known. Draw BE so that the angle ABE
is equal to the angle DBG, and let BE
meet AG in E.
Then the triangles A BE, DBG are
equiangular, and therefore
AB:AE=BD:DC,
or AB.DC= AE.BD. (1)
Again, to each of the equal angles
ABE, DBG add the angle EBD ;
then the angle ABD is equal to the angle EBC, and the
triangles ABD, EBC are equiangular ;
therefore BC : CE = BD: DA,
or AD.BC = CE.BD.
By adding (1) and (2), we obtain
AB.DC+AD.BC=AC. BD.
(2)
Now let AB, AC be two arcs terminating at A, the extremity
of the diameter. AD of a circle, and let
AG (= oc) be greater than AB (=/?;.
Suppose that (crd. AC) and (crd. AB)
are given : it is required to find
(crd. BC).
Join BD y CD.
Then, by the above theorem,
AC .BD = BC . AD + AB .CD.
Now AB, AC are given"; therefore BD = crd. (180° -AB)
and CD = crd. (180° — A C) are known. And AD is known.
Hence the remaining chord BC (crd. BC) is known.
280 TRIGONOMETRY
The equation in fact gives the formula,
[crd. («-£)}. (crd. 180°) = (crd. a) . [crd. (180°-j8)}
-(crd. /3).{crd. (180°-a)},
which is, of course, equivalent to
sin ($ — 0) = sin cos — cos sin 0, where a — 2 0, (3 = 20.
By means of this formula Ptolemy obtained
crd. 12° = crd. (72°-60°) = 12P 32' 36".
(8) Equivalent of sin 2 J0 = § (1— cos0).
But, in order to get the chords of smaller angles still, we
want a formula for finding the chord of half an arc when the
chord of the arc is given. This is the subject of Ptolemy's
next proposition.
Let BG be an arc of a circle with diameter AC, and let the
arc BC be bisected at D. Given (crd. BG), it is required to
find (crd. DC).
Draw DF perpendicular to AG,
and join AB, AD, BD, DC. Measure
A E along AG equal to AB, and join
BE.
Then shall FG be equal to EF, or
FG shall be half the difference be-
tween AC and AB.
For the triangles A BD, AED are
equal in all respects, since two sides
of the one are equal to two sides of the other and the included
angles BAD, EAD, standing on equal arcs, are equal.
Therefore ED = BD = DC,
and the right-angled triangles DEF, DCF are equal in all
respects, whence EF = FG, or CF = ±(AC-AB).
Now AC.CF=CD\
whence (crd. CDf = \AC (AG-AB)
= i(crd. 180 o ).{(crd.l80 o )-(erd.l80 o -£(7)}.
This is, of course, equivalent to the formula
sin 2 i<9 = |(1 -cos 6).
PTOLEMY'S tiYNTAXIS 281
i
By successively applying this formula, Ptolemy obtained
(crd. 6°), (crd. 3°) and finally (crd. 1|°) = 1* 34' 15" and
(crd. |°) = OP 47' 8". But we want a table going by half-
degrees, and hence two more things are necessary ; we have to
get a value for (crd. 1°) lying between (crd. 1|°) and (crd. f °),
and we have to obtain an addition formula enabling us when
(crd. a) is given to find {crd. (a + J°)}, and so on.
(e) Equivalent of cos (0 + (p) = cos 6 cos — sin sin 0.
To find the addition formula. Suppose AD is the diameter
of a circle, and AB, BG two arcs. Given (crd. AB) and
(crd. BG), to find (crd. AG). Draw the diameter BOE, and
join CE, GD, DE, BD.
Now, (crd. AB) being known,
(crd. BD) is known, and therefore
also (crd. DE), which is equal to
(crd. AB) ; and, (crd. BG) being
known, (crd. CE) is known.
And, by Ptolemy's theorem,
BD . CE = BG . DE+BE. GD.
The diameter BE and all the chords in this equation except
GD being given, we can find CD or crd. (180° — A C). We have
in fact
(crd. 180°) . {crd. (180° -AC)}
= {crd. (180° - AB) \.{cvd. (180° -BC)} -(crd. AB). (crd. BC);
thus crd. (180° — AC) and therefore (crd. AC) is known.
If AB = 2 0, BG = 2 0, the result is equivalent to
cos (0 + (f>) = cos cos — sin 6 sin 0.
(() Method of interpolation based on formula
sin oc /sin (3 < oi/fi (where \ tt > oc > ft).
Lastly we have to find (crd. 1°), having given (crd. 1J°) and
(crd. |°).
Ptolemy uses an ingenious method of interpolation based on
a proposition already assumed as known by Aristarchus.
If AB, BG be unequal chords in a circle, BG being the
282
TRIGONOMETRY
Now
greater, then shall the ratio of CB to BA be less than the
ratio of the arc CB to the arc BA.
Let BD bisect the angle ABC, jneeting AC in E and
the circumference in D. The arcs
AD, DC are then equal, and so are
the chords AD, DC. Also CE>EA
(since CB:BA = CE:EA).
Draw DF perpendicular to AC ;
then AD>DE>DF, so that the
circle with centre D and radius DE
will meet DA in G and DF produced
in#.
FE: EA = AFED : AAED
< (sector HED) : (sector GED)
< IFDEiAEDA.
Componendo, FA : AE < Z FDA : Z ^D^.
Doubling the antecedents, we have
CA:AE < LCDA-.LADE,
and, separando, CE:EA < Z CDE: Z EDA ;
therefore (since CB:BA = CE:EA)
CB.BA < ICDBUBDA
< (arc CB): (arc BA),
i. e. (crd. CB) : (crd. 5,4) < (arc CB) : (arc BA ).
[This is of course equivalent to sin oc : sin fi < a : /3, where
i7i->a>/3.] ,
It follows (1) that (crd. 1°) : (crd. j°)< 1 :|,
and (2) that (crd. li°) : (crd. 1°) < If : 1.
That is, -f . (crd. |°) > (crd. 1°) > | . (crd. 1|°).
But (crd. |°) = OP 47' 8", so that f (crd. |°) = IP 2' 50"
nearly (actually IP 2' SOf") ;
and (crd. lj°) = 1* 34' 15", so that |(crd. lf°) = IP ^ 50".
Since, then, (crd. 1°) is both less and greater than a length
which only differs inappreciably from IP 2' 50", we may say
that (crd. 1°) — \P 2' 50" as nearly as possible.
PTOLEMY'S SYNTAX Iti 283
(77) Table of Chords.
From this Ptolemy deduces that (crd. §°) is very nearly
0/' 31' 25", and by the aid of the above propositions he is in
a position to complete his Table of Chords for arcs subtending
angles increasing from J ° to 180° by steps of -|°; in other
words, a Table of Sines for angles from -|° to 90° by steps
of* .
(6) Further use of proportional increase.
Ptolemy carries further the principle of proportional in-
crease as a method of finding approximately the chords of
arcs containing an odd number of minutes between r and 30'.
Opposite each chord in the Table he enters in a third column
3 X oth of the excess of that chord over the one before, i.e. the
chord of the arc containing 30' less than the chord in question.
For example (crd. 2-|°) is stated in the second column of the
Table as 2P 37' 4". The excess of (crd. 2|°) over (crd. 2°) in the
Table is OP 31' 24"; ^th of this is OP Y 2" 48'", which is
therefore the amount entered in the third column opposite
(crd. 2^°). Accordingly, if we want (crd. 2° 25'), we take
(crd. 2°) or 2P 5' 40" and add 25 times OP l'2"48'"; or we
take (crd. 2|°) or 2P 37' 4" and subtract 5 times OP Y 2" 48'".
Ptolemy adds that if, by using the approximation for 1° and
J°, we gradually accumulate an error, we can check the calcu-
lation by comparing the chord with that of other related arcs,
e.g. the double, or the supplement (the difference between the
arc and the semicircle).
Some particular results obtained from the Table may be
mentioned. Since (crd. 1°) = 1 P 2' 50", the whole circumference
= 360 (IP 2' 50"), nearly, and, the length of the diameter
being 120*>, the value of n is 3 (1 +^_ + _5o_ ) _ 3 + ^8_ + _|o_
which is the value used later by Ptolemy and is equivalent to
3-14166... Again, a/3 = 2 sin 60° and, 2 (crd. 120°) being
equal to 2 (103? 55' 23"), we have V3 = ^ (103 + f£ + dnta)
43 55 23
= 1 +— .] + _ = 1-7320509,
60 60 2 60 3
which is correct to 6 places of decimals. Speaking generally,
284
TRIGONOMETRY
the sines obtained from Ptolemy's Table are correct to 5
places.
(l) Plane trigonometry in effect used.
There are other cases in Ptolemy in which plane trigono-
metry is in effect used, e.g. in the determination of the
eccentricity of the sun's orbit. 1 Suppose that AGBD is
the eccentric circle with centre 0,
and A B, GD are chords at right
angles through E, the centre of the
earth. To find OE. The arc BG
is known (= a, say) as also the arc
GA (=P). If BF be the chord
parallel to CD, and GG the chord
parallel to A B, and if iV, P be the
middle points of the arcs BF, GG,
Ptolemy finds (1) the arc BF
(= oc + /3- 180°), then the chord BF,
crd. (a +/3-180 ), then the half of it, (2) the arc GG
— arc (a + /3— 2/3) or arc (a — /?), then the chord GG, and
lastly half of it. He then adds the squares on the half-
chords, i.e. he obtains
0# 2 = i{crd. ( a + 0-18O)} 2 + f{crd.(a-/3)} 2 ,
that is, 0E*/r* = cos 2 J (oc + /3) + sin 2 § (a - 0).
He proceeds to obtain the angle OEG from its sine OR/ OE,
which he expresses as a chord of double the angle in the
circle on OE as diameter in relation to that diameter.
Spherical trigonometry : formulae in solution of
spherical triangles.
In spherical trigonometry, as already stated, Ptolemy
obtains everything that he wants by using the one funda-
mental proposition known as ' Menelaus's theorem ' applied
to the sphere (Menelaus III. 1), of which he gives a proof
following that given by Menelaus of the first case taken in
his proposition. Where Ptolemy has occasion for other pro-
positions of Menelaus's Sphaerica, e.g. III. 2 and 3, he does
1 Ptolemy, Syntaxis, iii. 4, vol. i, pp. 234-7.
PTOLEMY'S SYNTAX IS
285
not quote those propositions, as he might have done, but proves
them afresh by means of Menelaus's theorem. 1 The appli-
cation of the theorem in other cases gives in effect the
following different formulae belonging to the solution of
a spherical triangle ABC right-angled at C, viz.
sin a = sin c sin A,
tan a = sin b tan A,
cos c — cos a cos b,
tan b = tan c cos A.
One illustration of Ptolemy's procedure will be sufficient. 2
Let HAH' be the horizon, PEZH the meridian circle, EE'
the equator, ZZ' the ecliptic, F an
equinoctial point. Let EE', ZZ'
cut the horizon in A } B. Let P be
the pole, and let the great circle
through P, B cut the equator at C.
Now let it be required to find the
time which the arc FB of the ecliptic
takes to rise ; this time will be
measured by the arc FA of the
equator. (Ptolemy has previously found the length of the
arcs BC, the declination, and FC, the right ascension, of B,
I. 14, 16.)
By Menelaus's theorem applied to the arcs AE', E'P cut by
the arcs A H', PC which also intersect one another in B.
crd.2PH' crd.2PB crd. 2CA
that is,
crd. 2 H 'E f
sin PH'
crd. 2 BC ' cvd. 2 AE'
sin PB sin CA
sin H'E' ~~ sin BC sin AE'
Now sin PH' = cos H'E', sinPB^cosBC, and smAE'=l;
therefore cot H'E'= cot BC . sin CA ,
in other words, in the triangle ABC right-angled at C,
cot A — cot a sin b,
or tana = sin b tan A.
1 Syntaxis, vol. i, p. 169 and pp. 126-7 respectively.
2 A, vol. i, pp. 121-2.
286 TRIGONOMETRY
Thus AC is found, and therefore FC-AC or FA.
The lengths of BG, FG are found in I. 14, 16 by the same
method, the four intersecting great circles used in the figure
being in that case the equator EE', the ecliptic ZZ\ the great
circle PBCP' through the poles, and the great circle PKLP'
passing through the poles of both the ecliptic and the equator.
In this case the two arcs PL, AE f are cut by the intersecting
great circles PC, FK, and Menelaus's theorem gives (1)
sin PL _ sin OP sin BF ■
sin KL ~ sirTBG ' sin FK'
But sinPZ=l, sin KL = sin BFG, sinCP=l, sinPZ = l,
and it follows that
sin BG= sin BF sin BFC,
corresponding to the formula for a triangle right-angled at C,
sin a = sin c sin A.
(2) We have
sin PK sin PB sin GF
sin KL ' sin BG sin FL
and sin PK = cos KL = cos BFC, sin PB = cos BG, sin FL = 1 ,
so that tan BG = sin GF tan BFG,
corresponding to the formula
tan a = sin b tan A.
While, therefore, Ptolemy's method implicitly gives the
formulae for the solution of right-angled triangles above
quoted, he does not speak of right-angled triangles at all, but
only of arcs of intersecting great circles. The advantage
from his point of view is that he works in sines and cosines
only, avoiding tangents as such, and therefore he requires
tables of only one trigonometrical ratio, namely the sine (or,
as he has it, the chord of the double arc).
The Analcmma.
Two other works of Ptolemy should be mentioned here.
The first is the Analemma. The object of this is to explain
a method of representing on one plane the different points
THE ANALEMMA OF PTOLEMY 287
and arcs of the heavenly sphere by means of orthogonal
projection upon three planes mutually at right angles, the
meridian, the horizon, and the ' prime vertical '. The definite
problem attacked is that of showing the position of the sun at
any given time of the day, and the use of the method and
of the instruments described in the book by Ptolemy was
connected with the construction of sundials, as we learn from
Vitruvius. 1 There was 'another dvdXrj/i/ia besides that of
Ptolemy ; the author of it was Diodorus of Alexandria, a con-
temporary of Caesar and Cicero (' Diodorus, famed among the
makers of gnomons, tell me the time ! ' says the Anthology 2 ),
and Pappus wrote a commentary upon it in which, as he tells
us, 3 he used the conchoid in order to trisect an angle, a problem
evidently required in the Analemma in order to divide any
arc of a circle into six equal parts (hours). The word
dpdXrjizfxa evidently means ' taking up ' (* Aufnahme ') in the
sense of ' making a graphic representation ' of something, in
this case the representation on a plane of parts of the heavenly
sphere. Only a few fragments remain of the Greek text of
the Analemma of Ptolemy; these are contained in a palimpsest
(Ambros. Gr. L. 99 sup., now 491) attributed to the seventh
century but probably earlier. Besides this, we have a trans-
lation by William of Moerbeke from an Arabic version.
This Latin translation was edited with a valuable commentary
by the indefatigable Commandinus (Rome, 1562); but it is
now available in William of Moerbeke's own words, Heiberg
having edited it from Cod. Vaticanus Ottobon. lat. 1850 of the
thirteenth century (written in William's own hand), and in-
cluded it with the Greek fragments (so far as they exist) in
parallel columns in vol. ii of Ptolemy's works (Teubner, 1907).
The figure is referred to three fixed planes (1) the meridian,
(2) the horizon, (3) the prime vertical; these planes are the
planes of the three circles APZB, ACB, ZQG respectively
shown in the diagram below. Three other great circles are
used, one of which, the equator with pole P, is fixed ; the
other two are movable and were called by special names;
the first is the circle represented by any position of the circle
of the horizon as it revolves round G0G r as diameter (GSM in
1 Vitruvius, De architect, ix. 4. 2 Anth. Palat, xiv. 139.
8 Pappus, iv, p. 246, 1.
288
TRIGONOMETRY
the diagram is one position of it, coinciding with the equator),
and it was called eKrrjfjiopos kvkXo? (' the circle in six parts ')
because the highest point of it above the horizon corresponds
to the lapse of six hours ; the second, called the hour-circle, is
the circle represented by any position, as BSQA, of the circle
of the horizon as it revolves round B A as axis.
The problem is, as above stated, to find the position of the
sun at a given hour of the day. In order to illustrate
the method, it is sufficient, with A. v. Braunmuhl, 1 to take the
simplest case where the sun is on the equator, i.e. at one of
the equinoctial points, so that the hectemoron circle coincides
with the equator.
Let S be the position of the sun, lying on the equator MSC,
P the pole, MZA the meridian, BOA the horizon, BSQA the
hour-circle, and let the vertical great circle ZSV be drawn
through S, and the vertical great circle ZQC through Z the
zenith and G the east-point.
We are given the arc SO = 90° — t, where t is the hour-
angle, and the arc MB = 90° — (p, where <p is the elevation of
the pole ; and we have to find the arcs SV (the sun's altitude),
M>
o/
\/f\
^ \K
/S>)\
T "'\ >f
G
/ E \
/V: \
vV
H
<0 /° /
VC, the 'ascensional difference', SQ and QC. Ptolemy, in
fact, practically determines the position of 8 in terms of
certain spherical coordinates.
Draw the perpendiculars, SF to the plane of the meridian,
SH to that of the horizon, and SE to the plane of the prime
1 Braunmiilil, Gesch. der Trigonometrie, i, pp. 12, 13.
THE ANALEMMA OF PTOLEMY
289
vertical ; and draw FG perpendicular to BA, and ET to OZ.
Join HG, and we have FG = SH, GH = FS = ET.
We now represent SF in a separate figure (for clearness'
sake, as Ptolemy uses only one figure), where B'Z' A' corre-
sponds to BZA, P' to P and O'M' to OM. Set off the arc
P'S' equal to 08 (= 90° -t), and draw S'F' perpendicular
to O'M'. Then S'M'= 8M, and S'F r = SF; it is as if in the
original figure we had turned the quadrant MSG round MO
till it coincided with the meridian circle.
In the two figures draw IFK, I'F'K' parallel to BA, B'A\
and LFG, L'F'G' parallel to OZ, O'Z'.
Then (1) arc Zl — arc ZS = arc (90° — SV), because if we
turn the quadrant ZSV about ZO till it coincides with the
s ' Z
meridian, S falls on /, and V on B. It follows that the
required arc SV — arc B'l' in the second figure.
(2) To find the arc VC, set off G'X (in the second figure)
along G'F' equal to FS or F'S', and draw O'X through to
meet the circle in X'. Then arc ^'X'=arc VC; for it is as if
we had turned the quadrant BVG about BO till it coincided
with the meridian, when (since G'X = FS = GH) H would
coincide with X and V with X'. Therefore 5Fis also equal
to B'X'.
(3) To find QG or ZQ, set off along TF' in the second figure
T'Y equal to F'S\ and draw O'Y through to Y' on the circle.
Then arc B'Y' = arc QG: for it is as if we turned the prime
vertical ZQG about ZO till it coincided with the meridian,
when (since T'Y=S'F'= TE) E would fall on 7, the radius
OEQ on O'YY' and Q on Y'.
(4) Lastly, arc BS = arc BL = arc B'U, because 8, L are
u
1523.2
290 TRIGONOMETRY
both in the plane LSHG at right angles to the meridian ;
therefore arc SQ — arc UZ\
Hence all four arcs SV, VC, QC, QS are represented in the
auxiliary figure in one plane.
So far the procedure amounts to a method of grajMcally
constructing the arcs required as parts of an auxiliary circle
in one plane. But Ptolemy makes it clear that practical
calculation followed on the basis of the figure. 1 The lines
used in the construction are SF= sint (where the radius =1),
FT=0Fsin(f), FG = OF sin (90° -0), and this was fully
realized by Ptolemy. Thus he shows how to calculate the
arc SZ, the zenith distance (= d, say) or its complement 8V,
the height of the sun (= h, say), in the following way. He
says in effect: Since G is known, and Z F / 0'G / = 90° — 0, the
ratios O'F' : F'T and O'F' : O'T' are known.
*
0'F f T)
[In fact 7^7777-, = — , . n - E — — rr, where D is the diameter
L O'T' crd. (180° -20)'
of the sphere.]
Next, since the arc MS or M'S' is known [ = £], and there-
fore the arc P'S' [= 90°-t], the ratio of O'F' to D is known
[in fact 0'F'/D= {crd. (lS0-2t)}/2D.
It follows from these two results that
/= crd.(l 8 0°-2Q erd . QO _ '
2D x Yn
Lastly, the arc SV (= h) being equal to B'I\ the angle h is
equal to the angle 0'1'T in the triangle FQ'T'. And in this
triangle O'F, the radius, is known, while O'T' has been found ;
and we have therefore
O'T' crd.(2/<) crd. (180°- 2t) crd. (180°- 26) M
grp = — j = jj ^ , from above.
[In other words, sin h — cos t cos ; or, if u = SC = 90° — /,
sin h = sin u cos 0, the formula for finding sin h in the right-
angled spherical triangle SVC]
For the azimuth o> (arc BV — arc B'X'), the figure gives
xg' s'F' _ s'F' err 1
tan a) _ G , Q , -jTy,- , F ■ rr - tan t. ffi ^ ,
1 See Zeuthen in Bibh'otheca mathematics, h, 1900, pp. 23-7.
THE ANALEMMA OF PTOLEMY 291
or tan VG = tan SC cos SGV in the right-angled spherical
triangle SVG.
Thirdly,
tan QZ = tan Z Y = ^p = ^p • ^r, = tan < . -— ;
that is, 7 i^r-r = — — tj-^ , which is Menelaus, Sphaerica,
' tan#if sin #¥ z
III. 3, applied to the right-angled spherical triangles ZBQ,
MBS with the angle B common.
Zeuthen points out that later in the same treatise Ptolemy
finds the arc 2oc described above the horizon by a star of
given declination #', by a procedure equivalent to the formula
cos a = tan 8' tan 0,
and this is the same formula which, as we have seen,
Hipparchus must in effect have used in his Commentary on
the Phaenomena of Eudoxus and Aratus.
Lastly, with regard to the calculations of the height h and
the azimuth co in the general case where the sun's declination
is 8', Zeuthen has shown that they may be expressed by the
formulae
sin h = (cos 8' cos t — sin 8' tan 0) cos 0,
cos 8' sin t
and tana) =
k r + (cos 8' cos t — sin 8' tan 6) sin 6
cos
cos 8 / sin £
or
sin 8' cos + cos 8* cos £ sin
The statement therefore of A. v. Braunmtihl 1 that the
Indians were the first to utilize the method of projection
contained in the Analemma for actual trigonometrical calcu-
lations with the help of the Table of Chords or Sines requires
modification in so far as the Greeks at all events showed the
way to such use of the figure. Whether^the practical applica-
tion of the method of the Analemma for what is equivalent
to the solution of spherical triangles goes back as far as
Hipparchus is not certain ; but it is quite likely that it does,
1 Braunmuhl, i, pp. 13, 14, 38-41.
U 2
292 TRIGONOMETRY
seeing that Diodorus wrote his Analcmma in the next cen-
tury. The other alternative source for Hipparchus's spherical
trigonometry is the Menelaus-theorem applied to the sphere,
on which alone Ptolemy, as we have seen, relies in his
Syntaxis. In any case the Table of Chords or Sines was in
full use in Hipparchus's works, for it is presupposed by either
method.
The Planisphaerium.
With the Analemma of Ptolemy is associated another
work of somewhat similar content, the Planisphaerium.
This again has only survived in a Latin translation from an
Arabic version made by one Maslama b. Ahmad al-Majriti, of
Cordova (born probably at Madrid, died 1007/8) ; the transla-
tion is now found to be, not by Rudolph of Bruges, but by
'Hermannus Secundus', whose pupil Rudolph was; it was
first published at Basel in 1536, and again edited, with com-
mentary, by Commandinus (Venice, 1558). It has been
re-edited from the manuscripts by Heiberg in vol. ii. of his
text of Ptolemy. The book is an explanation of the system
of projection known as stereographic, by which points on the
heavenly sphere are represented on the plane of the equator
by projection from one point, a pole ; Ptolemy naturally takes
the south pole as centre of projection, as it is th£ northern
hemisphere which he is concerned to represent on a plane.
Ptolemy is aware that the projections of all circles on the
sphere (great circles — other than those through the poles
which project into straight lines — and small circles either
parallel or not parallel to the equator) are likewise circles.
It is curious, however, that he does not give any general
proof of the fact, but is content to prove it of particular
circles, such as the ecliptic, the horizon, &c. This is remark-
able, because it is easy to show that, if a cone be described
with the pole as vertex and passing through any circle on the
sphere, i.e. a circular cone, in general oblique, with that circle
as base, the section of the cone by the plane of the equator
satisfies the criterion found for the ' subcontrary sections ' by
Apollonius at the beginning of his Conies, and is therefore a
circle. The fact that the method of stereographic projection is
so easily connected with the property of subcontrary sections
THE PLAN1SPEAERIUM OF PTOLEMY 293
of oblique circular cones has led to the conjecture that Apollo-
nius was the discoverer of the method. But Ptolemy makes no
mention of Apollonius, and all that we know is* that Synesius
of Gyrene (a pupil of Hypatia, and born about a.d. 365-370)
attributes the discovery of the method and its application to
Hipparchus ; it is curious that he does not mention Ptolemy's
treatise on the subject, but speaks of himself alone as having
perfected the theory. While Ptolemy is fully aware that
circles on the sphere become circles in the projection, he says
nothing about the other characteristic of this method of pro-
jection, namely that the angles on the sphere are represented
by equal angles on the projection.
We must content ourselves with the shortest allusion to
other works of Ptolemy. There are, in the first place, other
minor astronomical works as follows :
(1) $d(T€is anXavtov da-re pcou of which only Book II sur-
vives, (2) 'TTTodiareis t&v TrXaucofxeucou in two Books, the first
of which is extant in Greek, the second in Arabic only, (3) the
inscription in Canobus, (4) Ilpoxeipcoy kclvovcdv SiaTacns kcu
yjrr)(po(popia. All these are included in Heiberg's edition,
vol. ii.
The Optics.
Ptolemy wrote an Optics in five Books, which was trans-
lated from an Arabic version into Latin ' in the twelfth
century by a certain Admiral Eugenius Siculus * ; Book I,
however, and the end of Book V are wanting. Books I, II
were physical, and dealt with generalities ; in Book III
Ptolemy takes up the theory of mirrors, Book IV deals with
concave and composite mirrors, and Book V with refraction.
The theoretical portion would suggest that the author was
not very proficient in geometry. Many questions are solved
incorrectly owing to the assumption of a principle which is
clearly false, namely that ' the image of a point on a mirror is
at the point of concurrence of two lines, one of which is drawn
from the luminous point to the centre of curvature of the
mirror, while the other is the line from the eye to the point
1 See G. Govi, L'ottica di Claudio Tolomeo di Euyenio Ammiraglio dA
Skilia, ... Torino, 1884; and particulars in G. Loria. Le scienze e*atte
nelV antica Grecia, pp. 570, 571.
294 TRIGONOMETRY
on the mirror where the reflection takes place ' ; Ptolemy uses
the principle to solve various special cases of the following
problem (depending in general on a biquadratic equation and
now known as the problem of Alhazen), ' Given a reflecting
surface, the position of a luminous point, and the position
of a point through which the reflected ray is required to pass,
to find the point on the mirror where the reflection will take
place.' Book V is the most .interesting, because it seems to
be the first attempt at a theory of refraction. It contains
many details of experiments with different media, air, glass,
and water, and gives tables of angles of refraction (r) corre-
sponding to different angles of incidence (i) ; these are calcu-
lated on the supposition that r and i are connected by an
equation of the following form,
r = ai — bi 2 ,
where a, b are constants, which is worth noting as the first
recorded attempt to state a law of refraction.
The discovery of Ptolemy's Optics in the Arabic at once
made it clear that the work Be specvlis formerly attributed
to Ptolemy is not his, and it is now practically certain that it
is, at least in substance, by Heron. This is established partly
by internal evidence, e.g. the style and certain expressions
recalling others which are found in the same author's Auto-
mata and Dioptra, and partly by a quotation by Damianus
(On hypotheses in Optics, chap. 14) of a proposition proved by
' the mechanician Heron in his own Catoptrica ', which appears
in the work in question, but is not found in Ptolemy's Optics,
or in Euclid's. The proposition in question is to the effect
that of all broken straight lines from the eye to the mirror
and from that again to the object, that particular broken line
is shortest in which the two parts make equal angles with the
surface of the mirror; the inference is that, as nature does
nothing in vain, we must assume that, in reflection from a
mirror, the ray takes the shortest course, i.e. the angles of
incidence and reflection are equal. Except for the notice in
Damianus and a fragment in Olympiodorus l containing the
proof of the proposition, nothing remains of the Greek text ;
1 Olympiodorus on Aristotle, Meteor, iii. 2, ed. Ideler, ii, p. 96, ed.
Stiive, pp. 212. 5-213. 20.
THE OPTICS OF PTOLEMY 295
but the translation into Latin (now included in the Teubner
edition of Heron, ii, 1900, pp. 316-64), which was made by
William of Moerbeke in 1269, was evidently made from the
Greek and not from the Arabic, as is shown by Graecisms in
the translation.
A mechanical work, Tlepl poircou.
There are allusions in Simplicius 1 and elsewhere to a book
by Ptolemy of mechanical content, nepl poncou, on balancings
or turnings of the scale, in which Ptolemy maintained as
against Aristotle that air or water (e.g.) in their own ' place '
have no weight, and, when they are in their own ' place ', either
remain at rest or rotate simply, the tendency to go up or to
fall down being due to the desire of things which are not in
their own places to move to them. Ptolemy went so far as to
maintain that a bottle full of air was not only not heavier
than the same bottle empty (as Aristotle held), but actually
lighter when inflated than when empty. The same work is
apparently meant by the ' book on the elements ' mentioned
by Simplicius. 2 Suidas attributes to Ptolemy three Books of
Mechanica.
Simplicius 3 also mentions a single book, wepl o^aorao-ea)?,
1 On t&mension ', i. e. dimensions, in which Ptolemy tried to
show that the possible number of dimensions is limited to
three.
Attempt to prove the Parallel-Postulate.
Nor should we omit to notice Ptolemy's attempt to prove
the Parallel-Postulate. Ptolemy devoted a tract to this
subject, and Proclus 4 has given us the essentials of the argu-
ment used. Ptolemy gives, first, a proof of Eucl. I. 28, and
then an attempted proof of I. 29, from which he deduces
Postulate 5.
1 Simplicius on Arist. De caelo, p. 710. 14, Heib. (Ptoleniv, ed. Heib.,
vol. ii, p. 263).
2 lb., p. 20. 10 sq.
3 lb., p. 9. 21 sq., (Ptolemy, ed. Heib., vol. ii, p. 265).
4 Proclus on Eucl. I, pp. 362. 14 sq., 365. 7-367. 27 (Ptolemy, ed. Heib.,
vol. ii, pp. 266-70).
296 TRIGONOMETRY
I. To prove I. 28, Ptolemy takes two straight lines AB, CD,
and a transversal EFGH. We have to prove that, if the sum
of the angles BFG, FGD is equal to two right angles, the
straight lines AB, CD are parallel, i.e. non-secant.
Since AFG is the supplement of BFG, and FGC of FGD, it
follows that the sum of the angles AFG, FGC is also equal to
two right angles. ,
Now suppose, if possible, that FB, GD, making the sum of
the angles BFG y FGD equal to two right angles, meet at K ;
then similarly FA, GC making the sum of the angles AFG,
FGC equal to two right angles must also meet, say at L.
[Ptolemy would have done better to point out that not
only are the two sums equal but the angles .themselves are
equal in pairs, i.e. AFG to FGD and FGC to BFG, and we can
therefore take the triangle KFG and apply it to FG on the other
side so that the sides FK, GK may lie along GC, FA respec-
tively, in which case GC, FA will meet at the point where
K falls.]
Consequently the straight lines LABK, LCDK enclose a
space : which is impossible.
It follows that AB, CD cannot meet in either direction ;
they are therefore parallel.
II. To prove I. 29, Ptolemy takes two parallel lines AB,
CD and the transversal FG, and argues thus. It is required
to prove that Z AFG + Z CGF = two right angles.
For, if the sum is not equal to two right angles, it must be
either (1) greater or (2) less.
(1) If it is greater, the sum of the angles on the other side,
BFG, FGD, which are the supplements of the first pair of
angles, must be less than two right angles.
But AF, CG are no more parallel than FB, GD, so that, if
FG makes one pair of angles AFG, FGC together greater than
PTOLEMY ON THE PARALLEL-POSTULATE 297
two right angles, it miist also make the other pair BFG, FGD
together greater than two right angles.
But the latter pair of angles were proved less than two
right angles : which is impossible.
Therefore the sum of the angles AFG, FGC cannot be
greater than two right angles.
(2) Similarly we can show that the sum of the two angles
AFG, FGC cannot be less than two right angles.
Therefore Z AFG + Z CGF = two right angles.
[The fallacy here lies in the inference which I have marked
by italics. When Ptolemy says that AF, CG are no more
parallel than FB, GD, he is in effect assuming that through
any one point only one parallel can be drawn to a given straight
line, which is an equivalent for the very Postulate he is
endeavouring to prove. The alternative Postulate is known
as ' Playfair's axiom ', but it is of ancient origin, since it is
distinctly enunciated in Proclus's note on Eucl. I. 31.]
III. Post. 5 is now deduced, thus.
Suppose that the straight lines making with a transversal
angles the sum of which is less than two right angles do not
meet on the side on which those angles are.
Then, a fortiori, they will not meet on the other side on
which are the angles the sum of which is greater than two
right angles. [This is enforced by a supplementary proposi-
tion showing that, if the lines met on that side, Eucl. I. 16
would be contradicted.]
Hence the straight lines cannot meet in either direction :
they are therefore parallel.
But in that case the angles made with the transversal are
equal to two right angles : which contradicts the assumption.
Therefore the straight lines will meet.
XVIII
MENSURATION: HERON OF ALEXANDRIA
Controversies as to Heron's date.
The vexed question of Heron's date has perhaps called
forth as much discussion as any doubtful point in the history
of mathematics. In the early stages of the controversy much
was made of the supposed relation of Heron to Ctesibius.
The Belopoeica of Heron has, in the best manuscript, the
heading "Hponvos Krr\cn^iov BeXoirouKa, and from this, coupled
with an expression used by an anonymous Byzantine writer
of the tenth century, 6 'AcrKprjvb? Kr-qcrifiios 6 Tov'AXtgavSpim
"Hpavos KadrjyrjTrjs, ' Ctesibius of Ascra, the teacher of Heron
of Alexandria ', it was inferred that Heron was a pupil of
Ctesibius. The question then was, when did Ctesibius live ?
Martin took him to be a certain barber of that name who
lived in the time of Ptolemy Euergetes II, that is, Ptolemy VII,
called Physcon (died 117 B.C.), and who is said to have made
an improved water-organ l ; Martin therefore placed Heron at
the beginning of the first century (say 126-50) B.C. But
Philon of Byzantium, who repeatedly mentions Ctesibius by
name, says that the first mechanicians (rex^Tai) had the
great advantage of being under kings who loved fame and
supported the arts. 2 This description applies much better
to Ptolemy II Philadelphus (285-247) and Ptolemy III Euer-
getes I (247-222). It is more probable, therefore, that Ctesibius
was the mechanician Ctesibius who is mentioned by Athenaeus
as having made an elegant drinking-horn in the time of
Ptolemy Philadelphus 3 ; a pupil then of Ctesibius would
probably belong to the end of the third and the beginning of
the second century B.C. But in truth we cannot safely con-
clude that Heron was an immediate pupil of Ctesibius. The
Byzantine writer probably only inferred this from the title
1 Athenaeus, Deipno-Soph. iv. c. 75, p. 174 b-e: cf. Vitruvius, x. 9, 13.
2 Philon, Mechan. Synt., p. 50. 38, ed. Schone.
3 Athenaeus, xi. c. 97, p. 497 b-e.
CONTROVERSIES AS TO HERON'S DATE 299
above quoted ; the title, however, in itself need not imply
more than that Heron's work was a new edition of a similar
work by Ctesibius,and the Ktyio-l^lov may even have been added
by some well-read editor who knew both works and desired to
indicate that the greater part of the contents of Heron's work
was due to Ctesibius. One manuscript h.3LS f 'Hpcovos 'A\e£ai>-
Specos BtXoTrouKd, which corresponds to the titles of the other
works of Heron and is therefore more likely to be genuine.
The discovery of the Greek text of the Metrica by R. Schone
in 1896 made it possible to fix with certainty an upper limit.
In that work there are a number of allusions to Archimedes,
three references to the yapiov aTTOTop,r\ of Apollonius, and
two to ' the (books) about straight lines (chords) in a circle '
(SeStLKTcu 8e kv rots rrepl tcou kv kvkXco zvOeicov). Now, although
the first beginnings of trigonometry may go back as far as
Apollonius, we know of no work giving an actual Table of
Chords earlier than that of Hipparchus. We get, therefore,
at once the date 1 50 B. c. or thereabouts as the terminus post
quern. A terminus ante quern is furnished by the date of the
composition of Pappus's Collection ; for Pappus alludes to, and
draws upon, the works of Heron. As Pappus was writing in
the reign of Diocletian (a.d. 284-305), it follows that Heron
could not be much later than, say, a.d. 250. In speaking of
the solutions by ' the old geometers ' (ol naXaiol yeco/jLtTpcu) of
the problem of finding the two mean proportionals, Pappus may
seem at first sight to include Heron along with Eratosthenes,
Nicomedes and Philon in that designation, and it has been
argued, on this basis, that Heron lived long before Pappus.
But a close examination of the passage 1 shows that this is
by no means necessary. The relevant words are as follows :
1 The ancient geometers were not able to solve the problem
of the two straight lines [the problem of finding two mean
proportionals to them] by ordinary geometrical methods, since
the problem is by nature <; solid "... but by attacking it with
mechanical means they managed, in a wonderful way, to
reduce the question to a practical and convenient construction,
as may be seen in the Mesolabon of Eratosthenes and in the
mechanics of Philon and Heron . . . Nicomedes also solved it
by means of the cochloid curve, with which he also trisected
an angle.'
1 Pappus, iii, pp. 54-6.
300 HERON OF ALEXANDRIA
Pappus goes on to say that he will give four solutions, one
of which is his own ; the first, second, and third he describes
as those of Eratosthenes, Nicomedes and Heron. But in the
earlier sentence he mentions Philon along with Heron, and we
know from Eutocius that Heron's solution is practically the
same as Philon's. Hence we may conclude that by the third
solution Pappus really meant Philon's, and that he only men-
tioned Heron's Mechanics because it was a convenient place in
which to find the same solution.
Another argument has been based on the fact that the
extracts from Heron's Mechanics given at the end of Pappus's
Book VIII, as we have it, are introduced by the author with
a complaint that the copies of Heron's works in which he
found them were in many respects corrupt, having lost both
beginning and end. 1 But the extracts appear to have been
added, not by Pappus, but by some later writer, and the
argument accordingly falls to the ground.
The limits of date being then, say, 150 B.C. to A. D* 250, our
only course is to try to define, as well as possible, the relation
in time between Heron and the other mathematicians who
come, roughly, within the same limits. This method has led
one of the most recent writers on the subject (Tittel 2 ) to
place Heron not much later than 100 B.C., while another, 3
relying almost entirely on a comparison between passages in
Ptolemy and Heron, arrives at the very different conclusion
that Heron was later than Ptolemy and belonged in fact to
the second century a.d.
In view of the difference between these results, it will be
convenient to summarize the evidence relied on to establish
the earlier date, and to consider how far it is or is not con-
clusive against the later. We begin with the relation of
Heron to Philon. Philon is supposed to come not more than
a generation later than Ctesibius, because it would appear that
machines for throwing projectiles constructed by Ctesibius
and Philon respectively were both available at one time for
inspection by experts on the subject 4 ; it is inferred that
1 Pappus, viii, p. 1116. 4-7.
2 Art. ' Heron von Alexandreia' in Pauly-Wissowa's Real-Encyclopddie
dei class. Altertumstvissenschaft, vol. 8. 1, 1912.
3 I. Hammer-Jensen in Hermes, vol. 48, 1913, pp. 224-35.
4 Philon, Mech. Spit, iv, pp. 68. 1, 72. 36.
CONTROVERSIES AS TO HERON'S DATE 301
Philon's date cannot be later than the end of the second
century B.C. (If Ctesibius flourished before 247 B.C. the argu-
ment would apparently suggest rather the beginning than the
end of the second century.) Next, Heron is supposed to have
been a younger contemporary of Philon, the grounds being
the following. (1) Heron mentions a ' stationary-automaton'
representation by Philon of the Nauplius-story, 1 and this is
identified by Tittel with a representation of the same story by
some contemporary of Heron's (ol kocO' f)/xd$ 2 ). But a careful
perusal of the whole passage seems to me rather to suggest
that the latter representation was not Philon's, and that
Philon was included by Heron among the { ancient ' auto-
maton-makers, and not among his contemporaries." (2) Another
argument adduced to show that Philon was contemporary
1 Heron, Autom., pp. 404. 11-408. 9. 2 lb., p. 412. 13.
3 The relevant remarks of Heron are as follows. (1) He says that he
has found no arrangements of 'stationary automata' better or more
instructive than those described by Philon of Byzantium (p. 404. 11).
As an instance he mentions Philon's setting of the Nauplius-story, in
which he found everything good except two things (a) the mechanism
for the appearance of Athene, which was too difficult (epycodeo-Tepov), and
(b) the absence of an incident promised by Philon in his description,
namely the falling of a thunderbolt on Ajax with a sound of thunder
accompanying it (pp. 404. 15-408. 9). This latter incident Heron could
not find anywhere in Philon, though he had consulted a great number
of copies of his work. He continues (p. 408. 9-13) that we are not to
suppose that he is running down Philon or charging him with not being
capable of carrying out what he promised. On the contrary, the omission
was probably due to a slip of memory, for it is easy enough to make
stage-thunder (he proceeds to show how to do it). But the rest of
Philon's arrangements seemed to him satisfactory, and this, he says, is
why he has not ignored Philon's work : ' for I think that my readers will
get the most benefit if they are shown, first what has been well said b}^
the ancients and then, separately from this, what the ancients overlooked
or what in their work needed improvement ' (pp. 408. 22-410. 6). (2) The
next chapter (pp. 410. 7-412. 2) explains generally the sort of thing the
automaton-picture has to show, and Heron says he will give one example
which»he regards as the best. Then (3), after drawing a contrast between
the simpler pictures made by ' the ancients ', which involved three different
movements only, and the contemporary (ol *a0' fjpas) representations of
interesting stories by means of more numerous and varied movements
(p. 412. 3-15), he proceeds to describe a setting of the Nauplius-story.
This is the representation which Tittel identifies with Philon's. But it
is to be observed that the description includes that of the episode of the
thunderbolt striking Ajax (c.30, pp. 448. 1-452. 7) which Heron expressly
says that Philon omitted. Further, the mechanism for the appearance
of Athene described in c. 29 is clearly not Philon's 'more difficult'
arrangement, but the simpler device described (pp. 404. 18-408. 5) as
possible and preferable to Philon's (cf. Heron, vol. i, ed. Schmidt, pp.
Ixviii-lxix).
302 HERON OF ALEXANDRIA
with Heron is the fact that Philon has some criticisms of
details of construction of projectile- throwers which are found
in Heron, whence it is inferred that Philon had Heron's work
specifically in view. But if Heron's BeXoirouKd was based on
the work of Ctesibius, it is equally possible that Philon may
be referring to Ctesibius.
A difficulty in the way of the earlier date is the relation in
which Heron stands to Posidonius. In Heron's Mechanics,
i. 24, there is a definition of ' centre of gravity ' which is
attributed by Heron to ' Posidonius a Stoic '. But this can
hardly be Posidonius of Apamea, Cicero's teacher, because the
next sentence in Heron, stating a distinction drawn by Archi-
medes in connexion with this definition, seems to imply that
the Posidonius referred to lived before Archimedes. But the
Definitions of Heron do contain definitions of geometrical
notions which are put down by Proclus to Posidonius of
Apamea or Rhodes, and, in particular, definitions of ' figure '
and of 'parallels'. Now Posidonius lived from 135 to 51 B.C.,
and the supporters of the earlier date for Heron can only
suggest that either Posidonius was not the first to give these
definitions, or alternatively, if he was, and if they were
included in Heron's Definitions by Heron himself and not by
some later editor, all that this obliges us to admit is that
Heron cannot have lived before the first century B. c.
Again, if Heron lived at the beginning of the first cen-
tury B.C., it is remarkable that he is nowhere mentioned by
Vitruvius. The De architectural was apparently brought out
in 14 B.C. and in the preface to Book VII Vitruvius gives
a list of authorities on machinationes from whom he made
extracts. The list contains twelve names and has every
appearance of being scrupulously complete ; but, while it
includes Archytas (second), Archimedes (third), Ctesibius
(fourth), and Philon of Byzantium (sixth), it does not men-
tion Heron. Nor is it possible to establish interdependence
between Heron and Vitruvius ; the differences seem, on the
whole, to be more numerous than the resemblances. A few of
the differences may be mentioned. Vitruvius uses 3 as the
value of 7r, whereas Heron always uses the Archimedean value
3f. Both writers make extracts from the Aristotelian
Mrj^avLKa 7rpofi\rj /Accra, but their selections are different. The
CONTROVERSIES AS TO HERON'S DATE 303
machines used by the two for the same purpose frequently
differ in details ; e. g. in Vitru vius's hodometer a pebble drops
into a box at the end of each Roman mile. 1 while in Heron's
the distance completed is marked by a pointer. 2 It is indeed
pointed out that the water-organ of Heron is in many respects
more primitive than that of Vitruvius ; but, as the instru-
ments are altogether different, this can scarcely be said to
prove anything.
On the other hand, there are points of contact between
certain propositions of Heron and of the Roman agrimen-
sores. Columella, about a.d. 62, gave certain measurements of
plane figures which agree with the formulae used by Heron,
notably those for the equilateral triangle, the regular hexagon
(in this case not only the formula but the actual figures agree
with Heron's) and the segment of a circle which is less than
a semicircle, the formula in the last case being
where s is the chord and h the height of the segment. Here
there might seem to be dependence, one way or the other ;
but the possibility is not excluded that the two writers may
merely have drawn from a common source ; for Heron, in
giving the formula for the area of the segment of a circle,
states that it was the formula used by ' the more accurate
investigators' (ol ccKpL^icrrepou e ^77777 /core?). 3
We have, lastly, to consider the relation between Ptolemy
and Heron. If Heron lived about 100 B.C., he was 200 years
earlier than Ptolemy (a.d. 100—178). The argument used to
prove that Ptolemy came some time after Heron is based on
a passage of Proclus where Ptolemy is said to have remarked
on the untrustworthiness of the method in vogue among the
' more ancient ' writers of measuring the apparent diameter of
the sun by means of water-clocks. 4 Hipparchus, says Pro-
clus, used his dioptra for the purpose, and Ptolemy followed
him. Proclus proceeds :
' Let us then set out here not only the observations of
the ancients but also the construction of the dioptra of
1 Vitruvius, x. 14. 2 Heron, Dioptra, c. 34.
3 Heron, Metrica, i. 31, p. 74. 21.
4 Proclus, Hypotyposis, pp. 120. 9-15, 124. 7-26.
304 HERON OF ALEXANDRIA
Hipparchus. And first we will show how we can measure an
interval of time by means of the regular efflux of water,
a procedure which was explained by Heron the mechanician
in his treatise on water-clocks.'
Theon of Alexandria has a passage to a similar effect. 1 He
first says that the most ancient mathematicians contrived
a vessel which would let water flow out uniformly through a
small aperture at the bottom, and then adds at the end, almost
in the same words as Proclus uses, that Heron showed how
this is managed in the first book of his work on water-
clocks. Theon's account is from Pappus's Commentary on
the Syntaxis, and this is also Proclus's source, as is shown by
the fact that Proclus gives a drawing of the water-clock
which appears to have been lost in Theon's transcription from
Pappus, but which Pappus must have reproduced from the
work of Heron. Tittel infers that Heron must have ranked
as one of the ' more ancient ' writers as compared with
Ptolemy. But this again does not seem to be a necessary
inference. No doubt Heron's work was a convenient place to
refer to for a description of a water-clock, but it does not
necessarily follow that Ptolemy was referring to Heron's
clock rather than some earlier form of the same instrument.
An entirely different conclusion from that of Tittel is
reached in the article ' Ptolemaios and Heron ' already alluded
to. 2 The arguments are shortly these. (1) Ptolemy says in
his Geography (c. 3) that his predecessors had only been able
to measure the distance between two places (as an arc of a
great circle on the earth's circumference) in the case where
the two places are on the same meridian. He claims that he
himself invented a way of doing this even in the case where
the two places are neither on the same meridian nor on the
same parallel circle, provided that the heights of the pole at
the two places respectively, and the angle between the great
circle passing through both and the meridian circle through
one of the places, are known. Now Heron in his Dioptra
deals with the problem of measuring the distance between
two places by means of the dioptra, and takes as an example
1 Theon, Comm. on the Syntaxis, Basel, 1538, pp. 261 sq. (quoted in
Proclus, Hypotyposis, ed. Manitius, pp. 309-11).
2 Hammer-Jensen, op. cit.
CONTROVERSIES AS TO HERONS DATE 305
the distance between Rome and Alexandria. 1 Unfortunately
the text is in places corrupt and deficient, so that the method
cannot be reconstructed in detail. But it involved the obser-
vation of the same lunar eclipse at Rome and Alexandria
respectively and the drawing of the analemma for Rome.
That is to say, the mathematical method w T hich Ptolemy
claims to have invented is spoken of by Heron as a thing
generally known to experts and not more remarkable than
other technical matters dealt with in the same book. Conse-
quently Heron must have been later than Ptolemy. (It is
right to add that some hold that the chapter of the Dioptra
in question is not germane to the subject of the treatise, and
was probably not written by Heron but interpolated by some
later editor ; if this is so, the argument based upon it falls to
the ground.) (2) The dioptra described in Heron's work is a
fine and accurate instrument, very much better than anything
Ptolemy had at his disposal. If Ptolemy had been aware of
its existence, it is highly unlikely that he would have taken
the trouble to make his separate and imperfect ' parallactic '
instrument, since it could easily have been grafted on to
Heron's dioptra. Not only, therefore, must Heron have been
later than Ptolemy but, seeing that the technique of instru-
ment-making had made such strides in the interval, he must
have been considerably later. (3) In his work irepl po-rrcov 2
Ptolemy, as we have seen, disputed the view of Aristotle that
air has weight even when surrounded by air. Aristotle
satisfied himself experimentally that a vessel full of air is
heavier than the same vessel empty ; Ptolemy, also by ex-
periment, convinced himself that the former is actually the
lighter. Ptolemy then extended his argument to water, and
held that water with water round it has no weight, and that
the diver, however deep he dives, does not feel the weight of
the water above him. Heron 3 asserts that water has no
appreciable weight and has no appreciable power of com-
pressing the air in a vessel inverted and forced down into
the water. In confirmation of this he cites the case of the
diver, who is not prevented from breathing when far below
1 Heron, Dioptra, c. 35 (vol. iii, pp. 302-6).
2 Simplicius on De caelo, p. 710. 14, Heib. (Ptolemy, vol. ii, p. 263).
3 Heron, Pneum«tica, i. Pref. (vol. i, p. 22. 14 sq.).
1523.2 X.
306 HERON OF ALEXANDRIA
the surface. He then inquires what is the reason why the
diver is not oppressed though he has an unlimited weight of
water on his back. He accepts, therefore, the view of Ptolemy
as to the fact, however strange this may seem. But he is not
satisfied with the explanation given : ' Some say ', he goes on,
' it is because water in itself is uniformly heavy (/o-o/Sape? avro
Kad' avro) ' — this seems to be equivalent to Ptolemy's dictum
that water in water has no weight — ' but they give no ex-
planation whatever why divers . . .' He himself attempts an
explanation based on Archimedes. It is suggested, therefore,
that Heron's criticism is directed specifically against Ptolemy
and no one else. (4) It is suggested that the Dionysius to whom
Heron dedicated his Definitions is a certain Dionysius who
was r praefectus urbi at Rome in a.d. 301. The grounds are
these (a) Heron addresses Dionysius as Aiovvcrie XafX7rp6raTe,
where Xau-rrpoTaTo? obviously corresponds to the Latin clarissi-
rnus, a title which in the third century and under Diocletian
was not yet in common use. Further, this Dionysius was
curator aquarv/m and curator operum publicorum, so that he
was the sort of person who would have to do with the
engineers, architects and craftsmen for whom Heron wrote.
Lastly, he is mentioned in an inscription commemorating an
improvement of water supply and dedicated ' to Tiberinus,
father of all waters, and to the ancient inventors of marvel-
lous constructions ' (repertoribus admirabilium fabricarum
pritcis viris), an expression which is not found in any other
inscription, but which recalls the sort of tribute that Heron
frequently pays to his predecessors. This identification of the
two persons named Dionysius is an ingenious conjecture, but
the evidence is not such as to make it anything more. 1
The result of the whole investigation just summarized is to
place Heron in the third century A.D., and perhaps little, if
anything, earlier than Pappus. Heiberg accepts this conclu-
sion, 2 which may therefore, I suppose, be said to hold the field
for the present.
1 Dionysius was of course a very common name. Diophantus dedicated
his Arithmetica to a person of this name {rifuarare /uoi kiovixxu), whom he
praised for his ambition to learn the solutions of arithmetical problems.
This Dionysius must have lived in the second half of the third century
A. D., and if Heron also belonged to this time, is it not possible that
Heron's Dionysius was the same person?
2 Heron, vol. v, p. ix,
CONTROVERSIES AS TO HERON'S DATE 307
Heron was known as 6 'AXegavSpevs (e.g. by Pappus) or
6 nrf^aviKos (mechanicus), to distinguish him from other
persons of the same name ; Proclus and Damianus use the
latter title, while Pappus also speaks of ol nepl rbv "Hpoova
/j.r}\a^LK0L.
Character of works.
Heron was an almost encyclopaedic writer on mathematical
and physical subjects. Practical utility rather than theoreti-
cal completeness was the object aimed at; his environment in
Egypt no doubt accounts largely for this. His Metrica begins
with the old legend of the traditional origin of geometry in
Egypt, and in the Dioptra we find one of the very problems
which geometry was intended to solve, namely that of re-
establishing boundaries of lands when the flooding of the
Nile had destroyed the land -marks : ' When the boundaries
of an area have become obliterated to such an extent that
only two or three marks remain, in addition to a plan of the
area, to supply afresh the remaining marks.' 1 Heron makes
little or no claim to originality; he often quotes authorities,
but, in accordance with Greek practice, he more frequently
omits to do so, evidently without any idea of misleading any
one ; only when he has made what is in his opinion any
slight improvement on the methods of his predecessors does
he trouble to mention the fact, a habit which clearly indi-
cates that, except in these cases, he is simply giving the best
traditional methods in the form which seemed to him easiest
of comprehension and application. The Metrica seems to be
richest in definite references to the discoveries of prede-
cessors ; the names mentioned are Archimedes, Dionysodorus,
Eudoxus, Plato ; in the Dioptra Eratosthenes is quoted, and
in the introduction to the Gatoptrica Plato and Aristotle are
mentioned.
The practical utility of Heron's manuals being so great, it
was natural that they should have great vogue, and equally
natural that the most popular of them at any rate should be
re-edited, altered and added to by later writers ; this was
inevitable with books which, like the Elements of Euclid,
were in regular use in Greek, Byzantine, Roman, and Arabian
1 Heron, Dioptra, c. 25, p. 268. 17-19.
x 2
308 HERON OF ALEXANDRIA
education for centuries. The geometrical or mensurational
books in particular gave scope for expansion by multiplication
of examples, so that it is difficult to disentangle the genuine
Heron from the rest of the collections which have come down
to us under his name. Hultsch's considered criterion is as
follows : ' The Heron texts which have come down to our
time are authentic in so far as they bear the author's name
and have kept the original design and form of Heron's works,
but are unauthentic in so far as, being constantly in use for
practical purposes, they were repeatedly re-edited and, in the
course of re-editing, were rewritten with a view to the
particular needs of the time.'
List of Treatises.
Such of the works of Heron as have survived have reached
us in very different ways. Those which have come down in
the Greek are :
I. The Metrica, first discovered in 1896 in a manuscript
of the eleventh (or twelfth) century at Constantinople by
R. Schone and edited by his son, H. Scheme (Heronis Opera, iii,
Teubner, 1903).
II. On the Dioptra, edited in an Italian version by Venturi
in 1814 ; the Greek text was first brought out by A. J. H.
Vincent 1 in 1858, and the critical edition of it byH. Schone is
included in the Teubner vol. iii just mentioned.
III. The Pneumatica, in two Books, which appeared first in
a Latin translation by Commandinus, published after his
death in 1575; the Greek text was first edited by TheVenot
in Vetevum mathematicovum opera Graece et Latine edita
(Paris, 1693), and is now available in Heronis Opera, i (Teub-
ner, 1899), by W. Schmidt.
IV. On the art of constructing automata (irepl avTo/jiaTo-
iroLrjTLKrjs), or The automaton-theatre, first edited in an Italian
translation by B. Baldi in 1589 ; the Greek text was included
in TheVenot's Vet, math., and now forms part of Heronis
Opera, vol. i, by W. Schmidt.
V. Belopoe'ica (on the construction of engines of war), edited
1 Notices et extraits des manuscrits de la Biblioiheque impe'riale, xix, pt. 2,
pp. 157-337.
LIST OF TREATISES 309
by B. Baldi (Augsburg, 1616), Thevenot (Vet. math.), Kochly
and Riistow (1853) and by Wescher (Pollorcetique des Grecs,
1867, the first critical edition).
VI. The Cheirobalistra ('Hpcovos ^ipofiaWicrTpas KaracrKevrj
Kal <rv/i/j.€Tpca (?)), edited by V. Prou, Notices et extraits, xxvi. 2
(Paris, 1877).
VII. The geometrical works, Definitiones, Geometria, Geo-
daesia, Stereometrica I and II, Mensurae, Liber Geeponicus,
edited by Hultsch with Variae collectiones (Heronis Alexan-
drini geometrioorum et stereometricorum reliquiae, 1864).
This edition will now be replaced by that of Heiberg in the
Teubner collection (vols, iv, v), which contains much addi-
tional matter from the Constantinople manuscript referred to,
but omits the Liber Geeponicus (except a few extracts) and the
Geodaesia (which contains only a few extracts from the
Geometry of Heron).
Only fragments survive of the Greek text of the Mechanics
in three Books, which, however, is extant in the Arabic (now
edited, with German translation, in Heronis Opera, vol. ii,
by L. Nix and W. Schmidt, Teubner, 1901).
A smaller separate mechanical treatise, the BapovXKo?, is
quoted by Pappus. 1 The object of it was ' to move a given
weight by means of a given force ', and the machine consisted
of an arrangement of interacting toothed wheels with different
diameters.
At the end of the Dioptra is a description of a hodometer for
measuring distances traversed by a wheeled vehicle, a kind of
taxameter, likewise made of a combination of toothed wheels.
A work on Water-clocks (irepl vdpioov oapocrKOTrtioov) is men-
tioned in the Pneumatica as having contained four Books,
and is also alluded to by Pappus. 2 Fragments are preserved
in Proclus (Hypotyposis, chap. 4) and in Pappus's commentary
on Book V of Ptolemy's Syntaxis reproduced by Theon.
Of Heron's Commentary on Euclid's Elements only very
meagre fragments survive in Greek (Proclus), but a large
number of extracts are fortunately preserved in the Arabic
commentary of an-NairizI, edited (1) in the Latin version of
Gherard of Cremona by Curtze (Teubner, 1899), and (2) by
1 Pappus, viii, p. 1060. 5. 2 lb., p. 1026. 1.
310 HERON OF ALEXANDRIA
Besthorn and Heiberg {Codex Leidensis 399. 1, five parts of
which had appeared up to 1910). The commentary extended
as far as Elem. VIII. 27 at least.
The Catoptrica, as above remarked under Ptolemy, exists in
a Latin translation from the Greek, presumed to be by William
of Moerbeke, and is included in vol. ii of Heronis Opera,
edited, with introduction, by W. Schmidt.
Nothing is known of the Camarica (' on vaultings ') men-
tioned by Eutocius (on Archimedes, Sphere and Cylinder), the
Zygicc (balancings) associated by Pappus with the Automata, 1
or of a work on the use of the astrolabe mentioned in the
Fihrist.
We are in this work concerned with the treatises of mathe-
matical content, and therefore can leave out of account such
works as the Pneumatica, the Automata, and the Belopoe'ica.
The Pneumatica and Automata have, however, an interest to
the historian of physics in so far as they employ the force of
compressed air, water, or steam. In the Pneumatica the
reader will find such things as siphons, ' Heron's fountain ',
' penny-in-the-slot ' machines, a fire-engine, a water-organ, and
many arrangements employing the force of steam.
Geometry.
(a) Commentary on Euclid's Elements.
In giving an account of the geometry and mensuration
(or geodesy) of Heron it will be well, I think, to begin
with what relates to the elements, and first the Commen-
tary on Euclid's Elements, of which we possess a number
of extracts in an-Nairlzi and Proclus, enabling us to form
a general idea of the character of the work. Speaking
generally, Heron's comments do not appear to have contained
much that can be called important. They may be classified
as follows :
(1) A few general notes, e.g. that Heron would not admit
more than three axioms.
(2) Distinctions of a number of particular caves of Euclid's
propositions according as the figure is drawn in one way
or another.
1 Pappus, viii, p. 1024. 28.
GEOMETRY 311
Of this class are the different cases of I. 35, 36, III. 7, 8
(where the chords to be compared are drawn on different sides
of the diameter instead of on the same side), III. 12 (which is
not Euclid's at all but Heron's own, adding the case of
external to that of internal contact in III. 11 \ VI. 19 (where
the triangle in which an additional line is drawn is taken to
be the smaller of the two), VII. 19 (where the particular case
is given of three numbers in continued proportion instead of
four proportionals).
(3) Alternative proofs.
It appears to be Heron who first introduced the easy but
uninstructive semi-algebraical method of proving the proposi-
tions II. 2-10 which is now so popular. On this method the
propositions are proved ' without figures ' as consequences of
II. 1 corresponding to the algebraical formula
a (b + c + d + . . .) = ab + ac -f ad + . . .
Heron explains that it is not possible to prove II. 1 without
drawing a number of lines (i. e. without actually drawing the
rectangles), but that the following propositions up to II. 10
can be proved by merely drawing one line. He distinguishes
two varieties of the method, one by dissolutio, the other by
compositio, by which he seems to mean splitting -up of rect-
angles and squares and combination of them into others.
But in his proofs he sometimes combines the two varieties.
Alternative proofs are given (a) of some propositions of
Book III, namely III. 25 (placed after III. 30 and starting
from the arc instead of the chord), III. 10 (proved by means
of III. 9), III. 13 (a proof preceded by a lemma to the effect
that a straight line cannot meet a circle in more than two
points).
A class of alternative proof is (6) that which is intended to
meet a particular objection (eWracriy) which had been or might
be raised to Euclid's constructions. Thus in certain cases
Heron avoids producing a certain straight line, where Euclid
produces it, the object being to meet the objection of one who
should deny our right to assume that there is any space
available. Of this class are his proofs of I. 11, 20 and his
note on I. 16. Similarly in I. 48 he supposes the right-angled
312 HERON OF ALEXANDRIA
triangle which is constructed to be constructed on the same
side of the common side as the given triangle is.
A third class (c) is that which avoids reductio ad absurdum,
e.g. a direct proof of I. 19 (for which he requires and gives
a preliminary lemma) and of I. 25.
(4) Heron supplies certain converses of Euclid's propositions
e.g. of II. 12, 13 and VIII. 27.
(5) A few additions to, and extensions of, Euclid's propositions
are also found. Some are unimportant, e. g. the construction
of isosceles and scalene triangles in a note on I. 1 and the
construction of two tangents in III. 17. The most important
extension is that of III. 20 to the case where the angle at the
circumference is greater than a right angle, which gives an
easy way of proving the theorem of III. 22. Interesting also
are the notes on I. 37 (on I. 24 in Proclus), where Heron
proves that two triangles with two sides of the one equal
to two sides of the other and with the included angles supple-
mentary are equal in area, and compares the areas where the
sum of the included angles (one being supposed greater than
the other) is less or greater than two right angles, and on I. 47,
where there is a proof (depending on preliminary lemmas) of
the fact that, in the figure of Euclid's proposition (see next
page), the straight lines AL, BG, GE meet in a point. This
last proof is worth giving. First come the lemmas.
(1) If in a triangle ABG a straight line DE be drawn
parallel to the base BG cutting the sides AB, AC or those
sides produced in D, E, and if F be the
middle point of BG, then the straight line
AF (produced if necessary) will also bisect
DE. (HK is drawn through A parallel to
DE, and HDL, REM through D, E parallel
to AF meeting the base in L, M respec-
tively. Then the triangles ABF, AFC
between the same parallels are equal. So are the triangles
DBF, EFC. Therefore the differences, the triangles ADF,
AEF, are equal and so therefore are the parallelograms HF,
KF. Therefore LF = FM, or DG = GE.)
(2) is the converse of Eucl. 1. 43. If a parallelogram is
GEOMETRY
313
cut into four others ADGE, I)F, FGGB, GE, so that DF, GE
are equal, the common vertex G will lie on the diagonal AB.
Heron produces AG to meet GF in H, and then proves that
AHB is a straight line.
Since DF, GE are equal, so are
the triangles D GF, EGG. A dding
the triangle GGF, we have the
triangles EGF, JDGF equal, and
DE, GF are parallel.
But (by I. 34, 29, 26) the tri-
angles AKE, GKD are congruent,
so that EK=KD ; and by lemma ( 1) it follows that CH=HF.
Now, in the triangles FHB, CHG, two sides (BF, FH and
GC, GH) and the included angles are equal ; therefore the
triangles are congruent, and the angles BHF, GHG are equal.
Add to each the angle GHF, and
Z BHF+ Z FHG = Z CHG + Z GHF = two right angles.
To prove his substantive proposition Heron draws AKL
perpendicular to BG, and joins EG meeting AK in M. Then
we have only to prove that BMG is a straight line.
Complete the parallelogram FAHG, and draw the diagonals
OA, FH meeting in F. Through M draw PQ, SR parallel
respectively to BA, AG.
314 HERON OF ALEXANDRIA
Now the triangles FAH, BAG are equal in all respects ;
therefore IHFA = I ABC
= Z CAR (since A K is at right angles to BG).
But, the diagonals of the rectangle FH cutting one another
in Y, we have FY = YA and L.HFA = LOAF;
therefore LOAF — AGAK, and OA is in a straight line
. with AKL.
Therefore, OM being the diagonal of SQ, SA — AQ. and, if
we add AM to each, FM = MH. .
Also, since EG is the diagonal of FN, FM — MN.
Therefore the parallelograms MH, MN are equal ; and
hence, by the preceding lemma, BMG is a straight line. Q.E.D.
(ft) The Definitions.
The elaborate collection of Definitions is dedicated to one
Dionysius in a preface to the following effect :
'In setting out for you a sketch, in the shortest possible
form, of the technical terms premised in the elements of
geometry, I shall take as my point of departure, and shall
base my whole arrangement upon, the teaching of Euclid, the
author of the elements of theoretical geometry ; for by this
means I think that I shall give you a good general under-
standing not only of Euclid's doctrine but of many other
works in the domain of geometiy. I shall begin then with
the 'point!
He then proceeds to the definitions of the point, the line,
the different sorts of lines, straight, circular, ' curved ' and
' spiral-shaped ' (the Archimedean spiral and the cylindrical
helix), Defs. 1-7 ; surfaces, plane and not plane, solid body,
Defs. 8-11; angles and their different kinds, plane, solid,
rectilinear and not rectilinear, right, acute and obtuse angles,
Defs. 12-22; figure, boundaries of figure, varieties of figure,
plane, solid, composite (of homogeneous or non-homogeneous
parts) and incomposite, Defs. 23-6. The incomposite plane
figure is the circle, and definitions follow of its parts, segments
(which are composite of non-homogeneous parts), the semi-
circle, the a\jfis (less than a semicircle), and the segment
greater than a semicircle, angles in segments, the sector,
THE DEFINITIONS 315
' concave ' and ' convex ', lune, garland (these last two are
composite of homogeneous parts) and axe (ireXeKvs), bounded by-
four circular arcs, two concave and two convex, Defs. 2 7-38.
Rectilineal figures follow, the various kinds of triangles and
of quadrilaterals, the gnomon in a parallelogram, and the
gnomon in the more general sense of the figure which added
to a given figure makes the whole into a similar figure,
polygons, the parts of figures (side, diagonal, height of a
triangle), perpendicular, parallels, the three figures which will
fill up the space round a point, Defs. 39-73. Solid figures are
next classified according to the surfaces bounding them, and
lines on surfaces are divided into (1) simple and circular,
(2) mixed, like the conic and spiric curves, Defs. 74, 75. The
sphere is then defined, with its parts, and stated to be
the figure which, of all figures having the same surface, is the
greatest in content, Defs. 76-82. Next the cone, its different
species and its parts are taken up, with the distinction
between the three conies, the section of the acute-angled cone
(' by some also called ellipse ') and the sections of the right-
angled and obtuse-angled cones (also called 'parabola and
hyperbola), Defs. 83-94; the cylinder, a section in general,
the spire or tore in its three varieties, open, continuous (or
just closed) and ' crossing-itself ', which respectively have
sections possessing special properties, ' square rings ' which
are cut out of cylinders (i. e. presumably rings the cross-section
of which through the centre is two squares), and various other
figures cut out of spheres or mixed surfaces, Defs. 95-7 ;
rectilineal solid figures, pyramids, the five regular solids, the
semi-regular solids of Archimedes two of which (each with
fourteen faces) were known to Plato, Defs. 98-104; prisms
of different kinds, parallelepipeds, with the special varieties,
the cube, the beam, Bokos (length longer than breadth and
depth, which may be equal), the brick, ttXivOis (length less
than breadth and depth), the o-cprjvicrKos or /3co/jll(tkos with
length, breadth and depth unequal, Defs. 105-14.
Lastly come definitions of relations, equality of lines, sur-
faces, and solids respectively, similarity of figures, ' reciprocal
figures', Defs. 115-18; indefinite increase in magnitude,
parts (which must be homogeneous with the wholes, so that
e. g. the horn-like angle is not a part or submultiple of a right
316 HERON OF ALEXANDRIA
or any angle), multiples, Dels. 119-21 ; proportion in magni-
tudes, what magnitudes can have a ratio to one another,
magnitudes in the same ratio or magnitudes in proportion,
definition of greater ratio, Defs. 122-5; transformation of
ratios (componendo, separando, convertendo, altemando, in-
vertendo and ex aequali), Defs. 126-7 ; commensurable and
incommensurable magnitudes and straight lines, Defs. 128,
129. There follow two tables of measures, Defs. 130—2.
The Definitions are very valuable from the point of view of
the historian of mathematics, for they give the different alter-
native definitions of the fundamental conceptions; thus we
find the Archimedean ' definition ' of a straight line, other
definitions which we know from Proclus to be due to Apol-
lonius, others from Posidonius, and so on. No doubt the
collection may have been recast by some editor or editors
after Heron's time, but it seems, at least in substance, to go
back to Heron or earlier still. So far as it contains original
definitions of Posidonius, it cannot have been compiled earlier
than the first century B.C.; but its content seems to belong in
the main to the period before the Christian era. Heiberg
adds to his edition of the Definitions extracts from Heron's
Geometry, postulates and axioms from Euclid, extracts from
Geminus on the classification of mathematics, the principles
of geometry, &c, extracts from Proclus or some early collec-
tion of scholia on Euclid, and extracts from Anatolius and
Theon of Smyrna, which followed the actual definitions in the
manuscripts. These various additions were apparently collected
by some Byzantine editor, perhaps of the eleventh century.
Mensuration.
The Metrica, Geometrica, Stereometrica, Geodaesia,
Mensurae.
We now come to the mensuration of Heron. Of the
different works under this head the Metrica is the most
important from our point of view because it seems, more than
any of the others, to have preserved its original form. It is
also more fundamental in that it gives the theoretical basis of
the formulae used, and is not a mere application of rules to
particular examples. It is also more akin to theory in that it
MENSURATION 317
does not use concrete measures, but simple numbers or units
which may then in particular cases be taken to be feet, cubits,
or any other unit of measurement. Up to 1896, when a
manuscript of it was discovered by R. Schone at Constanti-
nople, it was only known by an allusion to it in Eutocius
(on Archimedes's Measurement of a Circle), who states that
the way to obtain an approximation to the square root of
a non-square number is shown by Heron in his Metrica, as
well as by Pappus, Theon, and others who had commented on
the Syntaxis of Ptolemy. 1 Tannery 2 had already in 1894
discovered a fragment of Heron's Metrica giving the particular
rule in a Paris manuscript of the thirteenth century contain-
ing Prolegomena to the Syntaxis compiled presumably from
the commentaries of Pappus and Theon. Another interesting
difference between the Metrica and the other works is that in
the former the Greek way of writing fractions (which is our
method) largely preponderates, the Egyptian form (which
expresses a fraction as the sum of diminishing submultiples)
being used . comparatively rarely, whereas the reverse is the
case in the other works.
In view of the greater authority of the Metrica, we shall
take it as the basis of our account of the mensuration, while
keeping the other works in view. It is desirable at the
outset to compare broadly the contents of the various collec-
tions. Book I of the Metrica contains the mensuration of
squares, rectangles and triangles (chaps. 1-9), parallel-trapezia,
rhombi, rhomboids and quadrilaterals with one angle right
(10-16), regular polygons from the equilateral triangle to the
regular dodecagon (17-25), a ring between two concentric
circles (26), segments of circles (27-33), an ellipse (34), a para-
bolic segment (35), the surfaces of a cylinder (36), an isosceles
cone (37), a sphere (38) and a segment of a sphere (39).
Book II gives the mensuration of certain solids, the solid
content of a cone (chap. 1), a cylinder (2), rectilinear solid
figures, a parallelepiped, a prism, a pyramid and a frustum,
&c. (3-8), a frustum of a cone (9, 10), a sphere and a segment
of a sphere (11, 12), a spire or tore (13), the section of a
cylinder measured in Archimedes's Method (14), and the solid
1 Archimedes, vol. iii, p. 232. 13-17.
2 Tannery, Memoires scientifiques, ii, 1912, pp. 447-54.
318 HERON OF ALEXANDRIA
formed by the intersection of two cylinders with axes at right
angles inscribed in a cube, also measured in the Method (15),
the five regular solids (16-19). Book III deals with the divi-
sion of figures into parts having given ratios to one another,
first plane figures (1-19), then solids, a pyramid, a cone and a
frustum, a sphere (20-3),
The Geometrla or Geometrumena is a collection based upon
Heron, but not his work in its present form. The addition of
a theorem due to Patricius 1 and a reference to him in the
Stereometrica (I. 22) suggest that Patricius edited both works,
but the date of Patricius is uncertain. Tannery identifies
him with a mathematical professor of the tenth century,
Nicephorus Patricius ; if this is correct, he would be contem-
porary with the Byzantine writer (erroneously called Heron)
who is known to have edited genuine works of Heron, and
indeed Patricius and the anonymous Byzantine might be one
and the same person. The mensuration in the Geometry has
reference almost entirely to the same figures as those
measured in Book I of the Metrica, the difference being that
in the Geometry (1) the rules are not explained but merely
applied to examples, (2) a large number of numerical illustra-
tions are given for each figure, (3) the Egyptian way of
writing fractions as the sum of submultiples is followed,
(4) lengths and areas are given in terms of particular
measures, and the calculations are lengthened by a consider-
able amount of conversion from one measure into another.
The first chapters (1-4) are of the nature of a general intro-
duction, including certain definitions and ending with a table
of measures. Chaps. 5-99, Hultsch ( = 5-20, 14, Heib.), though
for the most part corresponding in content to Metrica I,
seem to have been based on a different collection, because
chaps. 100-3 and 105 ( = 21, 1-25, 22, 3-24, Heib.) are clearly
modelled on the Metrica, and 101 is headed 'A definition
(really ' measurement ') of a "circle in another book of Heron \
Heiberg transfers to the Geometrica U considerable amount of
the content of the so-called Liber Geeponicus, a badly ordered
collection consisting to a large extent of extracts from the
other works. Thus it begins with 41 definitions identical
with the same number of the Definitiones. Some sections
1 Geometrica, 21 26 (vol. iv, p. 386. 23).
MENSURATION 319
Heiberg puts side by side with corresponding sections of the
Geometrica in parallel columns ; others he inserts in suitable
places ; sections 78. 79 contain two important problems in
indeterminate analysis (= Geom. 24, 1-2, Heib.). Heiberg
adds, from the Constantinople manuscript containing the
Metrica, eleven more sections (chap. 24, 3-13) containing
indeterminate problems, and other sections (chap. 24, 14-30 and
37-51) giving the mensuration, mainly, of figures inscribed in or
circumscribed to others, e.g. squares or circles in triangles,
circles in squares, circles about triangles, and lastly of circles
and segments of circles.
The Stereometriea I has at the beginning the title Elcra-
ycoyal rcov arepeo/ieTpovfievcoy "Hpo&vos but, like the Geometrica,
seems to have been edited by Patricius. Chaps. 1-40 give the
mensuration of the geometrical solid figures, the sphere, the
cone, the frustum of a cone, the obelisk with circular base,
the cylinder* the 'pillar', the cube, the arfy-qvio-Kos (also called
6vv£), the fietovpov irpoeo-Kapupevfievov, pyramids, and frusta.
Some portions of this section of the book go back to Heron ;
thus in the measurement of the sphere chap. 1 = Metrica
II. 11, and both here and elsewhere the ordinary form of
fractions appears. Chaps. 41-54 measure the contents of cer-
tain buildings or other constructions, e.g. a theatre, an amphi-
theatre, a swimming-bath, a well, a ship, a wine-butt, and
the like.
The second collection, Stereometriea II, appears to be of
Byzantine origin and contains similar matter to Stereometriea I,
parts of which are here repeated. Chap. 31 (27, Heib.) gives
the problem of Thales, to find the height of a pillar or a tree
by the measurement of shadows ; the last sections measure
various pyramids, a prism, a /3co/j.i<rKo$ (little altar).
The Geodaesia is not an independent work, but only con-
tains extracts from the Geometry; thus chaps. 1— 16 = Geom.
5-31, Hultsch ( = 5, 2-12, 32, Heib.); chaps. 17-19 give the
methods of finding, in any scalene triangle the sides of which
are given, the segments of the base made by the perpendicular
from the vertex, and of finding the area direct by the well-
known ' formula of Heron ' ; i.e. we have here the equivalent of
Metrica I. 5-8.
Lastly, the /xeTprjaeis, or Mensurae, was attributed to Heron
320 HERON OF ALEXANDRIA
in an Archimedes manuscript of the ninth century, but can-
not in its present form be due to Heron, although portions of
it have points of contact with the genuine works. Sects. 2-27
measure all sorts of objects, e.g. stones of different shapes,
a pillar, a tower, a theatre, a ship, a vault, a hippodrome ; but
sects. 28-35 measure geometrical figures, a circle and segments
of a circle (cf. Metrica I), and sects. 36-48 on spheres, segments
of spheres, pyramids, cones and frusta are closely connected
with Stereom. I and Metrica II ; sects. 49-59, giving the men-
suration of receptacles and plane figures of various shapes,
seem to have a different origin. We can now take up the
Contents of the Metrica.
Book I. Measurement of Areas.
The preface records the tradition that the first geometry
arose out of the practical necessity of measuring and dis-
tributing land (whence the name ' geometry '), after which
extension to three dimensions became necessary in order to
measure solid bodies. Heron then mentions Eudoxus and
Archimedes as pioneers in the discovery of difficult measure-
ments, Eudoxus having been the first to prove that a cylinder
is three times the cone on the same base and of equal height,
and that circles are to one another as the squares on their
diameters, while Archimedes first proved that the surface of
a sphere is equal to four times the area of a great circle in it,
and the volume two-thirds of the cylinder circumscribing it.
(a) Area of scalene triangle.
After the easy cases of the rectangle, the right-angled
triangle and the isosceles triangle, Heron gives two methods
of finding the area of a scalene triangle (acute-angled or
obtuse-angled) when the lengths of the three sides are given.
The first method is based on Eucl. II. 12 and 13. If a, b, c
be the sides of the triangle opposite to the angles A, B, C
respectively, Heron observes (chap. 4) that any angle, e.g. C, is
acute, right or obtuse according as c 2 < = or > a 2 + b 2 , and this
is the criterion determining which of the two propositions is
applicable. The method is directed to determining, first the
segments into which any side is divided by the perpendicular
AREA OF SCALANE TRIANGLE
321
from the opposite vertex, and thence the length of the per-
pendicular itself. We have, in the cases of the triangle acute-
angled at C and the triangle obtuse-angled at C respectively,
c 2 = a 2 + b 2 +2a.CD,
or GD = {(« 2 + /; 2 )-c 2 }/2^,
whence AD 2 (= b 2 — GD 2 ) is found, so that we know the area
(=ia.AD).
In the cases given in Metrica I. 5, 6 the sides are (14, 15, 13)
and (11, 13, 20) respectively, and AD is found to be rational
(=12). But of course both CD (or BD) and AD may be surds,
in which case Heron gives approximate values. Cf. Geom.
53, 54, Hultsch (15, 1-4, Heib.), where we have a triangle
in which a = 8, 6=4, c = 6, so that a 2 + b 2 — c 2 = 44 and
CD = 44/16 = 2|i. Thus AD'= 16-(2|J) 2 = 16-7| T V
= 8 i ¥ Te ' an d AD= -/(8J | ye) = 2§ i approximately, whence
the area = 4 x 2§ J = 11§. Heron then observes that we get
a nearer result still if we multiply AD 2 by (J a) 2 before
extracting the square root, for the area is then </(16 x 8 J f j-q)
or \/(135), which is very nearly 11| j 1 ^ Jt or Hff .
So in Metrica I. 9, where the triangle is 10, 8, 12 (10 being
the base), Heron finds the perpendicular to be a/63, but he
obtains the area as V ' {\AD 2 . BG 2 ). or */(1575), while observing
that we can, of course, take the approximation to a/63, or
7 "I i I ts> an d multiply it by half 10, obtaining 39 J | T X g as
the area.
Proof of the formula A = \/{s (s — a) (s — b) (s — c)}.
The second method is that known • as the ' formula of
Heron ', namely, in our notation, A = V { s (s — a) (s — b) (s — c) } .
The proof of the formula is given in Metrica I. 8 and also in
1523.2 Y
322
HERON OF ALEXANDRIA
chap. 30 of the Dioptra ; but it is now known (from Arabian
sources) that the proposition is due to Archimedes.
Let the sides of the triangle ABO be given in length.
Inscribe the circle DEF, and let be the centre.
Join AO, BO, GO, DO, EO, FO.
Then BC.0D=2AB0C,
GA.0E = 2AC0A,
AB.0F=2AA0B:
whence, by addition,
p.0D = 2 A ABO,
where p is the perimeter.
Produce CB to H, so that BH = AF.
Then, since AE = AF, BF = BD, and CE = OD, we have
CH=±p = s.
Therefore
CH.0D = AABC.
But CH.OD is the 'side' of the product GH 2 .OD 2 , i.e.
V(CH 2 . OD 2 ),
so that
(AABC) 2 = CH 2 .0D 2 .
PROOF OF THE FORMULA OF HERON' 323
Draw OL at right angles to OC cutting BC in K, and BL at
right angles to BC meeting OL in L. Join GL.
Then, since each of the angles COL, CBL is right, COBL is
a quadrilateral in a circle.
Therefore Z COB + Z 0X5 =25.
But LCOB + LAOF= 25, because 40, BO, CO bisect the
angles round 0, and the angles GOB, AOF are together equal
to the angles 400, 50.F, while the sum of all four angles
is equal to 45.
Consequently AA0F = Z CLB.
Therefore the right-angled triangles AOF, CLB are similar ;
therefore BC:BL = AF:F0
= BH-.OD,
and, alternately, CB:BH = BL: OD
= BK:KD;
whence, componendo, GH:HB — BD : DK.
It follows that
CH 2 :CH.HB = BD.DC : CD. DK
= BD.DC: OD 2 , since the angle COK is right.
Therefore (A ABC) 2 = CH 2 . OD 2 (from above)
= CH.HB. BD.DC
= s(s — a) (s - h) (s — e).
(/3) Method of approximating to the square root of
a non-square number.
It is a propos of the triangle 7, 8, 9 that Heron gives the
important statement of his method of approximating to the
value of a surd, which before the discovery of the passage
of the Metrica had been a subject of unlimited conjecture
as bearing on the question how Archimedes obtained his
approximations to VS.
In this case s = 12, s — a = 5, s — fr = 4, s — c = 3, so that
A = /(12 .5.4.3) = 7(720).
y2
324 HERON OF ALEXANDRIA
'Since', says Heron, 1 ' 720 has not its side rational, we can
obtain its side within a very small difference as follows. Since
the next succeeding square number is 729, which has 27 for
its side, divide 720 by 27. This gives 26|. Add 27 to this,
making 53§, and take half of this or 26 J J. The side of 720
will therefore be very nearly 26| §. In fact, if we multiply
26J§ by itself , the product is 720^, so that the difference (in
the square) is ^ .
' If we desire to make the difference still smaller than 3^-, we
shall take 720^ instead of 729 [or rather we should take
26J-| instead of 27], and by proceeding in the same way we
shall find that the resulting difference is much less than ■£$'
In other words, if we have a non-square number A, and a 2
is the nearest square number to it, so that A = a 2 + b, then we
have, as the first approximation to */A.
«!=!(«+ -); (D
for a second approximation we take
and so on. 2
1 Metrica, i. 8, pp. 18. 22-20. 5.
2 The method indicated by Heron was known to Barlaam and Nicolas
Rhabdas in th'e fourteenth century. The equivalent of it was used by
Luca Paciuolo (fifteenth -sixteenth century), and it was known to the other
Italian algebraists of the sixteenth century. Thus Luca Paciuolo gave
2\i 2^ and 2 T H 9 ^ 1 n as successive approximations to */6. He obtained
2 «. a 01 my-s
n—x, the second as 2| * rtl
2.2' l 2 . t\
tt-irir- The above rule - ives l(2+i) = 2|, i(|+-^)-2A,
" • 20 •
1 fiilj. l_liP\ — 9JL«JL
2 \20^ 48/ — ^1!>60-
The formula of Heron was again put forward, in modern times, by
Buzengeiger as a means of accounting for the Archimedean approxima-
tion to \/3, apparently without knowing its previous history. Bertrand
also stated it in a treatise on arithmetic (1853-). The method, too, by
which Oppermann and Alexeieff sought to account for Archimedes's
approximations is in reality the same. The latter method depends on
the formula
i(a + /3): v / (a3)-yW): a 2 ^-
Alexeieff separated A into two factors a , b , and pointed out that if. say.
"o> \/^4 >?? ,
then, i(a +b )>^/A> ** or -\ ° ,
the first as 2+ ^—7;, the second as 2| — J 2 ^, , and the third as
APPROXIMATIONS TO SURDS 325
a x = a + 7T-
Substituting in (1) the value a 2 + h for A, we obtain
b_
— 2a
Heron does not seem to have used this formula with a nejja-
tive sign, unless in Stereom. I. 33 (34, Hultsch), where \/(63)
and again, if h( a o + K) = a n 2A/(a + b Q ) = b lf
\( ih + b i )>VA> a ^ K
and so on.
Now suppose that, in Heron's formulae, we put a = X , A/a — x Q ,
0*1 = .A^, AjOL x = x\, and so on. We then have
A = i ' a + - ) = -J (A + x ), #, = — = , „ - or " ;
that is, Xj, ^! are, respectively, the arithmetic and harmonic means
between A , x ; X 2 , ic 2 are the arithmetic and harmonic means between
X y , x x , and so on, exactly as in AlexeiefFs formulae.
Let us now try to apply the method to Archimedes's case, «/3, and we
shall see to what extent it serves to give what we want. Suppose
we begin with 3 >y / 3 > 1. We then have
J(3 + l)>v / 3>3/^(3 + l), or 2> v / 3>:j,
and from this we derive successively
i ^ V ° ^ 7> 56^V d/ 9 7 > 10864 -^ V ° ^ 188IT*
But, if we start from f, obtained by the formula «+ ; <V (a 2 + b),
za + 1
we obtain the following approximations by excess,
1 (3. _Lii\ — 20 I (-Ail 4. 4_5\ _ 13Jl1
The second process then gives one of Archimedes's results, VWn bat
neither of the two processes gives the other, fff, directly. The latter
can, however, be obtained by using the formula that, if -<-, then
a ma + nc c
b mb + nd d
1? 14. • 265 t 97 1 168 U, 97 + 168 265
tor we can obtain ff§ from g^ and ^ftp thus : — - — ^= = -— - , or irom
oo + y / lOO
o, a * A1 11.97-7 1060 265 . 1UR1
f £ and 4 thus : n 56 _ 4 = "aTq" = f^S ' a s0 on " again ^falr can
be obtained from i&fM and ft thus : g§g±g - -Jg-g - ^ • .
The advantage of the method is that, as compared with that of con-
tinued fractions, it is a very rapid way of arriving at a close approxi-
mation, (jiinther has shown that the (m + l)th approximation obtained
by Heron's formula is the 2 m th obtained by continued fractions. ('Die
quadratischen Irrationalitaten der Alten und deren Entwickelungs-
methoden in Abhandlungen zur Geach. d. Math. iv. 18&2, pp. 83-6.)
326 HERON OF ALEXANDRIA
is given as approximately 8 — T V In Metrica I. 9, as we
have seen, \/(63) is given as 1\ \ § y 1 ^, which was doubtless
obtained from the formula (1) as
The above seems to be the only classical rule which has
been handed down for finding second and further approxi-
mations to the value of a surd. But, although Heron thus
shows how to obtain a second approximation, namely by
formula (2), he does not seem to make any direct use of
this method himself, and consequently the question how the
approximations closer than the first which are to be found in
his works were obtained still remains an open one.
(y) Quadrilaterals.
It is unnecessary to give in detail the methods of measuring
the areas of quadrilaterals (chaps. 11-16). Heron deals with
the following kinds, the parallel-trapezium (isosceles or non-
isosceles), the rhombus and rhomboid, and the quadrilateral
which has one angle right and in which the four sides have
given lengths. Heron points out that in the rhombus or
rhomboid, and in the general case of the quadrilateral, it is
necessary to know a diagonal as well as the four sides. The
mensuration in all the cases reduces to that of the rectangle
and triangle.
(8) The regular 'polygons with 3, 4, 5, 6, 7, 8, 9, 10, 11,
or 12 sides.
Beginning with the equilateral triangle (chap. 17), Heron
proves that, if a be the side and p the perpendicular from
a vertex on the opposite side, a 2 :p 2 = 4 : 3, whence
a 4 :^ 2 a 2 = 4:3 = 16:12,
so that a 4 :(AABC) 2 = 16:3,
and (AABC) 2 = ^a*. In the particular case taken a — 10
and A 2 = 1875, whence A = 43^ nearly.
Another method is to use an approximate value for \/3 in
the formula Vs . a 2 /4. This is what is done in the Geometrica
14 (10, Heib.), where we are told that the area is (§ + i 1 o)^ 2 ;
THE REGULAR POLYGONS 327
now 3 + xjt = || = i (f£)> so that the approximation used by
Heron for \/3 is here ff . For the side 10, the method gives
the same result as above, for §■§ . 100 = 43J.
The regular pentagon is next taken (chap. 18). Heron
premises the following lemma.
Let ABC be a right-angled triangle, with the angle A equal
to §i*. Produce AC to so that CO = AC
If now AO is divided in extreme and
mean ratio, A B is equal to the greater
segment. (For produce AB to D so that
,4D = AO, and join 50, DO. Then, since
J. 2)0 is isosceles and the angle at A—^R,
I ADO = AA0D = ±R, and, from the
equality of the triangles ABC, 0BC,
Z.A0B = LBA0 = fiS. It follows that
the triangle J.D0 is the isosceles triangle of Eucl. IV. 10, and
AD is divided in extreme and mean ratio in B.) Therefore,
says Heron, (BA+ACf = 5 AC 2 . [This is Eucl. XIII. 1.]
Now, since LB0C — %R, if BC be produced to E so that
CE = BC, BE subtends at an angle equal to ~R, and there-
fore BE is the side of a regular pentagon inscribed in the
circle with as centre and 0B as radius. (This circle also
passes through D, and BD is the side of a regular decagon in
the same circle.) If now B0 — AB = r, 0C = p, BE = a,
we have from above, (r + p) 2 — 5p 2 , whence, since V5 is
approximately J, we obtain approximately r = %p, and
la = %p, so that p — \a. Hence \pa — \o?, and the area
of the pentagon = fa 2 . Heron adds that, if we take a closer
approximation to a/5 than |, we shall obtain the area still
more exactly. In the Geometry 1 the formula is given as \^-a 2 .
The regular hexagon (chap. 19) is simply 6 times the
equilateral triangle with the same side. If A be the area
of the equilateral triangle with side a, Heron has proved
that A 2 = T 3 ga 4 (Metrica I. 17), hence (hexagon) 2 = $?-a\ If,
e.g. a = 10, (hexagon) 2 = 67500, and (hexagon) = 259 nearly.
In the Geometry 2 the formula is given as ^-a', while ' another
book' is quoted as giving 6(J + T ^)a 2 ; it is added that the
latter formula, obtained from the area of the triangle, (J + ^) a 2 ,
represents the more accurate procedure, and is fully set out by
1 Geom. 102 (21, 14, Heib.). 2 lb. 102 (21, 16, 17, Heib.).
328 HERON OF ALEXANDRIA
Heron. As a matter of fact, however, 6 (§ + jq) = -^ 3 - exactly,
and only the Metrica gives the more accurate calculation.
The regular heptagon.
Heron assumes (chap. 20) that, if a be the side and r the
radius of the circumscribing circle, a — |r, being approxi-
mately equal to the perpendicular from the centre of the
circle to the side of the regular hexagon inscribed in it (for |
is the approximate value of \ \^3). This theorem is quoted by
Jordanus Nemorarius (d. 1237) as an 'Indian rule'; he pro-
bably obtained it from Abu'l Wafa (940-98). The Metrica,
shows that it is of Greek origin, and, if Archimedes really
wrote a book on the heptagon in a circle, it may be clue to
him. If then p is the perpendicular from the centre of the
circle on the side (a) of the inscribed heptagon, r/(-|a) = 8/3-§
or 16/7, whence p 2 /(^a) 2 = -■iw'i an d 'p/\ a — (approxi-
mately) 14|/7 or 43/21. Consequently the area of the
heptagon = 7 . \pa — 7 . fftr — fi^ 2 -
The regular octagon, decagon and dodecagon.
In these cases (chaps. 21, 23, 25) Heron finds p by drawing
the perpendicular 00 from 0, the centre of the
circumscribed circle, on a side AB, and then making
the angle OAD equal to the angle AOD.
For the octagon,
I ADO = ±R, and p ^ Ja(l + V2) = Ja(l + f|)
or \a . f§ approximately.
For the decagon,
Z ADC = f R, and AD : DC =5:4 nearly (see preceding page) ;
hence AD : AC =5:3, and p = \a (§ + § ) — fa.
For the dodecagon,
Z ADC = | £, and p = |a (2 + V3) = \a (2 + J) = ^-a
approximately.
Accordingly A 8 = ^g-a 2 , A 10 = -^tt 2 , A2 = - 4 4 5 - a ' 2 > where a is
the side in each case.
The regular enneagon and hendecagon.
In these cases (chaps. 22, 24; the Table of Chords (i e.
THE REGULAR POLYGONS
329
presumably Hipparehus's Table) is appealed to. If AB be the
side (a) of an enneagon or hendecagon inscribed in a circle, AC
the diameter through A, we are told that the Table of Chords
gives § and ^ as the respective approximate values of the
ratio AB / AC. The angles subtended at the centre by the
side AB are 40° and 32 X 8 T ° respec-
tively, and Ptolemy's Table gives,
as the chords subtended by angles of
40° and 33° respectively, 41? 2' 33"
and 34P 4' 55" (expressed in 120th
parts of the diameter) ; Heron's
figures correspond to 40^ and 33^
36' respectively. For the enneagon
AG 2 = 9AB 2 , whence BC 2 =, SAB 2
or approximately 2££-AB 2 , and
BC = --§-a\ therefore (area of
enneagon)^ . AABC^-% 1 ^. For
the hendecagon AC 2 = % 5 -AB 2 and BC 2 = ^f-AB 2 , so that
BC = %r a > an d area of hendecagon = ^ . A ABC = --f-a 2 .
An ancient formula for the ratio between the side of any
regular polygon and the diameter of the circumscribing circle
is preserved in Geepon. 147 sq. (— Pseudo-Dioph. 23-41),
namely d n — n—. Now the ratio na n /d n tends to it as the
o
number ( n) of sides increases, and the formula indicates a time
when it was generally taken as = 3.
(e) The Circle.
Coming to the circle (Metrica I. 26) Heron uses Archi-
medes's value for 7r, namely - 2 T 2 -, making the circumference of
a circle *f-r and the area ^d 2 , where r is the radius and d the
diameter. It is here that he gives the more exact limits
for 77- which he says that Archimedes found in his work On
Plinthides and Cylinders, but which are not convenient for
calculations. The limits, as we have seen, are given in the
text as toVt^ 71 " < -w&stT' an< ^ with Tannery's alteration to
^Prir c 7T < -WAt 2- aue quite satisfactory. 1
See vol. i, pp. 232-3.
330 HERON OF ALEXANDRIA
(£) Segment of a circle.
According to Heron (Metrlca I. 30) the ancients measured
the area of a segment rather inaccurately, taking the area
to be \ (b + h) h, where b is the base and h the height. He
conjectures that it arose from taking n = 3, because, if we
apply the formula to the semicircle, the area becomes \ . 3 r l ,
where r is the radius. Those, he says (chap. 31), who have
investigated the area more accurately have added -^(V')'
to the above formula, making it ^(b + h)h + -£%(%b) % , and this
seems to correspond to the value 3^ for ir, since, when applied
to the semicircle, the formula gives \ (3r 2 + yT 2 ). He adds
that this formula should only be applied to segments of
a circle less than a semicircle, and not even to all of these, but
only in cases where b is not greater 'than Sh. Suppose e.g.
that b = 60, h = 1 ; in that case even J?(|&) 2 = jV 900 = 64f ,
which is greater even than the parallelogram with 60, 1 as
sides, which again is greater than the segment. Where there-
fore b > 3 h, he adopts another procedure.
This is exactly modelled on Archimedes's quadrature of
a segment of a parabola. Heron proves (Metrlca I. 27-29, 32)
that, if ADB be a segment of a circle, and D the middle point
of the arc, and if the arcs AD, DB be
similarly bisected at E, F,
A ADB < 4 (A AED + A DFB).
Similarly, if the same construction be
made for the segments AED, BFD, each
of them is less than 4 times the sum of the two small triangles
in the segments left over. It follows that
(area of segmt. ADB) > A ADB { 1 h £ + (£) 2 + ...}
> %AADB.
1 If therefore we measure the triangle, and add one-third of
it, we shall obtain the area of the segment as nearly as
possible.' That is, for segments in which b > Zh, Heron
takes the area to be equal to that of the parabolic segment
with the same base and height, or f bh.
In addition to these three formulae for S, the area of
a segment, there are yet others, namely
S = J (b + h) h (1 + 2t)> Mensurae 29,
X = i(b + h)h(l+ T \), „ 31.
SEGMENT OF A CIRCLE 331
The first of these formulae is applied to a segment greater
than a semicircle, the second to a segment less than a semi-
circle.
In the Metrica the area of a segment greater than a semi-
circle is obtained by subtracting the area of the complementary
segment from the area of the circle.
From the Geometrica 1 we find that the circumference of the
segment less than a semicircle was taken to be V(b 2 + 4/i 2 ) + \h
It
or alternatively V(b 2 + 4ft 2 ) + { V (b 2 + 4 h 2 ) — b}^-
(77) Ellipse, 'parabolic segment, surface of cylinder, right
cone, sphere and segment of sphere.
After the area of an ellipse (Metrica I. 34) and of a parabolic
segment (chap. 35), Heron gives the surface of a cylinder
(chap. 36) and a right cone (chap. 37) ; in both cases he unrolls
the surface on a plane so that the surface becomes that of a
parallelogram in the one case and a sector of a circle in the
other. For the surface of a sphere (chap. 38) and a segment of
it (chap. 39) he simply uses Archimedes's results.
Book. I ends with a hint how to measure irregular figures,
plane or not. If the figure is plane and bounded by an
irregular curve, neighbouring points are taken on the curve
such that, if they are joined in order, the contour of the
polygon so formed is not much different from the curve
itself, and the polygon is then measured by dividing it into
triangles. If the surface of an irregular solid figure is to be
found, you wrap round it pieces of very thin paper or cloth,
enough to cover it, and you then spread out the paper or
cloth and measure that.
Book II. Measurement of volumes.
The preface to Book II is interesting as showing how
vague the traditions about Archimedes had already become.
' After the measurement of surfaces, rectilinear or not, it is
proper to proceed to the solid bodies, the surfaces of which we
have already measured in the preceding book, surfaces plane
and spherical, conical and cylindrical, and irregular surfaces
as well. The methods of dealing with these solids are, in
1 Cf. Geom., 94, 95 (19. 2, 4, Heib.), 97. 4 (20. 7, Heib.).
332 HERON OF ALEXANDRIA
view of their surprising character, referred to Archimedes by
certain writers who give the traditional account of their
origin. But whether they belong to Archimedes or another,
it is necessary to give a sketch of these methods as well.'
The Book begins with generalities about figures all the
sections of which parallel to the base are equal to the base
and similarly situated, while the centres of the sections are on
a straight line through the centre of the base, which may be
either obliquely inclined or perpendicular to the base ; whether
the said straight line (' the axis ') is or is not perpendicular to
the base, the volume is equal to the product of the area of the
base and the perpendicular height of the top of the figure
from the base. The term ' height ' is thenceforward restricted
to the length of the perpendicular from the top of the figure
on the base.
(a) Cone, cylinder, parallelepiped (prism), pyramid, and
frustum.
II. 1-7 deal with a cone, a cylinder, a 'parallelepiped' (the
base of which is not restricted to the parallelogram but is in
the illustration given a regular hexagon, so that the figure is
more properly a prism with polygonal bases), a triangular
prism, a pyramid with base of any form, a frustum of a
triangular pyramid ; the figures are in general oblique,
(f3) Wedge-shaped solid (ficDfjLiorKos or crcprjvio-Kos).
II. 8 is a case which is perhaps worth giving. It is that of
a rectilineal solid, the base of which is a rectangle ABCD and
has opposite to it another rectangle EFGH, the sides of which
are respectively parallel but not necessarily proportional to
those of ABCD. Take A K equal to EF, and BL equal to FG.
Bisect BK, CL in V, W, and draw KRPU, VQOM parallel to
AD, and LQRN, WOPT parallel to AB. Join FK, GR, LG,
GU, BK
Then the solid is divided into (1) the parallelepiped with
AR, EG as oppqsite faces, (2) the prism with KL as base and
FG as the opposite edge, (3) the prism with NU as base and
GH as opposite edge, and (4) the pyramid with RLGU as base
and G as vertex. Let h be the ' height ' of the figure. Now
MEASUREMENT OF SOLIDS
333
the parallelepiped (1) is on AR as base and has height h ; the
prism (2) is equal to a parallelepiped on KQ as base and with
height h; the prism (3) is equal to a parallelepiped with NP
as base and height h; and finally the pyramid (4) is equal to
a parallelepiped of height h and one-third of RC as base.
Therefore the whole solid is equal to one parallelepiped
with height k and base equal to (AR + KQ + NP + RO + ^RO)
or AO + ^RO.
Now, if AB = a,BG= b, EF = c, FG = d,
AV = ±(a + c),AT = \{b + d),RQ = |(a-c), RP = i(b-d).
Therefore volume of solid
= {f(a + c)(& + d)+-5^(c&— c) (b-d)}h.
The solid in question is evidently the true ftcofiicrKos (' little
altar'), for the formula is used to calculate the content of
a fi<£>ii[(TKo$ in Stereom. II. 40 (68, Heib.) It is also, I think,
the <T(pr}VL(rKo? (' little wedge '), a measurement of which is
given in Stereom. I. 26 (25, Heib.) It is true that the second
term of the first factor ^ (a — c) (b — d) is there neglected,
perhaps because in the case taken {a — 7, b = 6, c = 5, d = 4)
this term (= •§■) is small compared with the other (=30). A
particular o-fy-qvio-Kos, in which either c = a or d = b, was
called 6vv§ ; the second term in the factor of the content
vanishes in this case, and, if e.g.c = a, the content is ^(b + d)ah.
Another ficofiio-Kos is measured in Stereom. I. 35 (34, Heib.),
where the solid is inaccurately called 'a pyramid oblong
(irepo/irJKi]^) and truncated (KoXovpos) or half-perfect'.
334 HERON OF ALEXANDRIA
The method is the same mutatis mutandis as that used in
II. 6 for the frustum of a pyramid with any triangle for base,
and it is applied in II. 9 to the case of a frustum of a pyramid
with a square base, the formula for which is
[{i(a + a')} 2 + Mi(«-a')i 2 ]*.
where a, a' are the sides of the larger and smaller bases
respectively, and h the height ; the expression is of course
easily reduced to J h(a 2 + aa' + a' 1 ).
(y) Frustum of cone, sphere, and segment thereof.
A frustum of a cone is next measured in two ways, (1) by
comparison with the corresponding frustum of the circum-
scribing pyramid with square base, (2) directly as the
difference between two cones (chaps. 9, 10). The volume of
the frustum of the cone is to that of the frustum of the
circumscribing pyramid as the area of the base of the cone to
that of the base of the pyramid ; i.e. the volume of the frus-
tum of the cone is \ it, or \\, times the above expression for
the frustum of the pyramid with a 2 , a' 2 as bases, and it
reduces to -^irh (a 2 + aa' + a' 2 ), where <x, a' are the diameters
of the two bases. For the sphere (chap. 11) Heron uses
Archimedes's proposition that the circumscribing cylinder is
1-| times the sphere, whence the volume of the sphere
= § . d . \^d 2 or If-cZ 3 ; for a segment of a sphere (chap. 12) he
likewise uses Archimedes's result {On the Sphere and Cylinder,
II. 4).
(8) Anchor-ring or tore.
The anchor-ring or tore is next measured (chap. 13) by
means of a proposition which Heron quotes from Dionyso-
dorus, and which is to the effect that, if a be the radius of either
circular section of the tore through the axis of revolution, and
c the distance of its centre from that axis,
ira 2 : etc = (volume of tore) : wc 2 . 2a
[whence volume of tore = 2n 2 ca 2 ~\. In the particular case
taken a = 6, c = 14, and Heron obtains, from the proportion
113^:84= 7:7392, F=9956f But he shows that he is
aware that the volume is the product of the area of the
MEASUREMENT OF SOLIDS 335
describing circle and the length of the path of its centre.
For, he says, since 1 4 is a radius (of the path of the centre),
28 is its diameter and 88 its circumference. c If then the tore
be straightened out and made into a cylinder, it will have 88
for its length, and the diameter of the base of the cylinder is
12; so that the solid content of the cylinder is, as we have
seen, 9956f ' (= 88 . i J . 144K
(e) The tivo special solids of Archimedes s ' Method '.
Chaps. 14, 15 give the measurement of the two remarkable
solids of Archimedes's Method, following Archimedes's results.
(() The Jive regular solids.
In chaps. 16-18 Heron measures the content of the five
regular solids after the cube. He has of course in each case
to find the perpendicular from the centre of the circumscrib-
ing sphere on any face. Let p be this perpendicular, a the
edge of the solid, r the radius of the circle circumscribing any
face. Then (1) for the tetrahedron
a 2 = 3r 2 , p 2 = a 2 — \a 2 = §a 2 .
(2) In the case of the octahedron, which is the sum of two
equal pyramids on a square base, the content is one-third
of that base multiplied by the diagonal of the figure,
i.e. J .a 2 , a/2 a or J a/2, a 3 ; in the case taken a = 7, and
Heron takes 10 as an approximation to \/(2 . 7 2 ) or a/98, the
result being J. 10.49 or 163|. (3) In the case of the icosa-
hedron Heron merely says that
p : a = 93 : 127 (the real value of the ratio is \ / 7 + 3 n/ 5 \ .
(4) In the case of the dodecahedron, Heron says that
~ n ali • , /25 + 11 \/5 . .„ / .
p:a = 9 : 8 (the true value is J / - -> and, if v5 is
put equal to J, Heron's ratio is readily obtained).
Book II ends with an allusion to the method attributed to
Archimedes for measuring the contents of irregular bodies by
immersing them in water and measuring the amount of fluid
displaced.
336 HERON OF ALEXANDRIA
Book III. Divisions of figures.
This book has much in common with Euclid's book On divi-
sions (of figures), the problem being to divide various figures,
plane or solid, by a straight line or plane into parts having
a given ratio. In III. 1-3 a triangle is divided into two parts
in a given ratio by a straight line (1) passing through a vertex,
(2) parallel to a side, (3) through any point on a side.
III. 4 is worth description : ' Given a triangle ABC, to cut
out of it a triangle DEF (where D, E, F are points on the
sides respectively) given in magnitude and such that the
triangles AEF, BFD, GET) may be equal in area.' Heron
assumes that, if D, E, F divide the sides so that
AF: FB = BD: DC = CE: EA,
the latter three triangles are equal in area.
He then has to find the value of
each of the three ratios which will
result in the triangle DEF having a
given area.
Join AD.
Since BD:CD = CE-.EA,
BC:CD= CA-.AE,
and AABC:AADC=AADC:AADE.
Also AABC: AABD = AADC: AEDC.
But (since the area of the triangle DEF is given) AEDC is
given, as well as AABC. Therefore AABD x A A DC is given.
Therefore, if A H be perpendicular to BC,
AH 2 .BD.DC is given;
therefore BD . DC is given, and, since BC is given, D is given
in position (we have to apply to BC a rectangle equal to
BD . DC and falling short by a square).
As an example Heron takes AB =13, BC =14, CA = 15,
ADEF = 24. AABC is then 84, and AH = 12.
Thus AEDC= 20, and AH 2 . BD. DC = 4 . 84 . 20 = 6720;
therefore BD .DC = 6720/144 or 46| (the text omits the §).
Therefore, says Heron, BD — 8 approximately. For 8 we
DIVISIONS OF FIGURES 337
should apparently have 8 J, since DC is immediately stated to
be 5J (not 6). That is, in solving the equation
x 2 -14&' + 46§ = 0,
which gives x — 7 ± V(2±), Heron apparently substituted 2 J or
f for 2§, thereby obtaining \\ as an approximation to the
surd.
(The lemma assumed in this proposition is easily proved.
Let m : n be the ratio AF: FB = BD : DC = GE-.EA.
Then AF — mc/(m + n), FB = nc/(m + n), GE — mb/(m + n),
EA = nb/(m + ri), &c.
Hence
AAFE/AABC = mn -aBDF/AABG = ACDE/AABG,
' (m + ny
and the triangles AFE, BDF, GDE are equal.
Pappus 1 has the proposition that the triangles A BG, DEF
have the same centre of gravity.)
Heron next shows how to divide a parallel-trapezium into
two parts in a given ratio by a straight line (l) through the
point of intersection of the non-parallel sides, (2) through a
given point on one of the parallel sides, (3) parallel to the
parallel sides, (4) through a point on one of the non-parallel
sides (III. 5-8). III. 9 shows how to divide the area of a
circle into parts which have a given ratio by means of an
inner circle with the same centre. For the problems begin-
ning with III. 10 Heron says that numerical calculation alone
no longer suffices, but geometrical methods must be applied.
Three problems are reduced to problems solved by Apollonius
in his treatise On cutting off an area. The first of these is
III. 10, to cut off from the angle of a triangle a given
proportion of the triangle by a straight line through a point
on the opposite side produced. III. 11. 12, 13 show how
to cut any quadrilateral into parts in a given ratio by a
straight line through a point (1) on a side (a) dividing the
side in the given ratio, (6) not so dividing it, (2) not on any
side, (a) in the case where the quadrilateral is a trapezium,
i.e. has two sides parallel, (b) in the case where it is not; the
last case (b) is reduced (like III. 10) to the ' cutting-off of an
1 Pappus, viii, pp. 1034-8. Cf. pp. 430-2 post.
1523 2 £
338 HERON OF ALEXANDRIA
area'. These propositions are ingenious and interesting.
III. 11 shall be given as a specimen.
Given any quadrilateral A BCD and a point E on the side
AD, to draw through E a straight line EF which shall cut
the quadrilateral into two parts in
the ratio of AE to ED. (We omit
the analysis.) Draw CG parallel
to DA to meet AB produced in G.
Join BE, and draw GH parallel
to BE meeting BC in H.
Join CE, EH, EG.
Then AGBE = AHBE and, adding A ABE to each, we have
AAGE = (quadrilateral ABHE).
Therefore (quadr. ABHE) : AC ED = A GAE: ACED
= AE:ED.
But (quadr. ABHE) and ACED are parts of the quadri-
lateral, and they leave over only the triangle EHC. We have
therefore only to divide A EHC in the same ratio AE-.ED by
the straight line EF. This is done by dividing HC at F in
the ratio AE: ED and joining EF.
The next proposition (III. 12) is easily reduced to this.
If AE : ED is not equal to the given ratio, let F divide AD
in the given ratio, and through F
draw FG dividing the quadri-
lateral in the given ratio (III. 11).
Join EG, and draw FH parallel
to EG. Let FH meet BC in H,
and join EH.
Then is EH the required straight
line through E dividing the quad-
rilateral in the given ratio.
For AFGE = AHGE. Add to each (quadr. GEDC).
Therefore (quadr. CGFD) = (quadr. CHED).
Therefore EH divides the quadrilateral in the given ratio,
just as FG does.
The case (III. 13) where E is not on a side of the quadri-
lateral [(2) above] takes two different forms according as the
DIVISIONS OF FIGURES
339
two opposite sides which the required straight line cuts are
(a) parallel or (b) not parallel. In the first case (a) the
problem reduces to drawing a straight line through E inter-
secting the parallel sides in points F, G such that BF+AG
is equal to a given length. In the second case (b) where
BG, AD are not parallel Heron supposes them to meet in H.
The angle at H is then given, and the area ABU. It is then
a question of cutting off from a triangle with vertex H a
triangle HFG of given area by a straight line drawn from E }
which is again a problem in Apollonius's Cutting-ojf of an
area. The auxiliary problem in case (a) is easily solved in
III. 16. Measure AH equal to the given length. Join BH
and bisect it at M. Then EM meets BG, AD in points such
that BF+ AG= the given length. For, by congruent triangles,
BF = GH.
The same problems are solved for the case of any polygon
in III. 14, 15. A sphere is then divided (III. 17) into segments
such that their surfaces are in a given ratio, by means of
Archimedes, On the Sphere and Cylinder, II. 3, just as, in
III. 23, Prop. 4 of the same Book is used to divide a sphere
into segments having their volumes in a given ratio.
III. 18 is interesting because it recalls an ingenious pro-
position in Euclid's book On Divisions. Heron's problem is
' To divide a given circle into three equal parts by two straight
z 2
340 HERON OF ALEXANDRIA
lines ', and he observes that, ' as the problem is clearly not
rational, we shall, for practical convenience, make the division,
as exactly as possible, in the follow-
ing way.' AB is the side of an
equilateral triangle inscribed in the
circle. Let CD be the parallel
diameter, the centre of the circle,
and join A0, BO, AD, DB. Then
shall the segment ABD be very
nearly one-third of the circle. For,
since AB is the side of an equi-
lateral triangle in the circle, the
sector OAEB is one-third of the
circle. And the triangle A OB forming part of the sector
is equal to the triangle ABB] therefore the segment AEB
r plus the triangle ABD is equal to one-third of the circle,
and the segment ABD only differs from this by the small
segment on BD as base, which may be neglected. Euclid's
proposition is to cut off one-third (or any fraction) of a circle
between two parallel chords (see vol. i, pp. 429-30).
III. 19 finds a point D within any triangle ABC such that
the triangles DBG, DO A, DAB are all equal ; and then Heron
passes to the division of solid figures.
The solid figures divided in a given ratio (besides the
sphere) are the pyramid with base of any form (III. 20),
the cone (III. 21) and the frustum of a cone (III. 22), the
cutting planes being parallel to the base in each case. These
problems involve the extraction of the cube root of a number
which is in general not an exact cube, and the point of
interest is Heron's method of approximating to the cube root
in such a case. Take the case of the cone, and suppose that
the portion to be cut off at the top is to the rest of the cone as
m to n. We have to find the ratio in which the height or the
edge is cut by the plane parallel to the base which cuts
the cone in the given ratio. The volume of a cone being
^irc 2 h, where c is the radius of the base and h the height,
we have to find the height of the cone the volume of which
is .kirc 2 h, and, as the height hf is to the radius c f of
m + 7b
its base as h is to c, we have simply to find h! where
DIVISIONS OF FIGURES 341
h' d /h? = 7ti/(m + n). Or, if we take the edges e, e' instead
of the heights, e' s /e 3 = m/(m + ri). In the case taken by-
Heron m : 71= 4 : 1, and e = 5. Consequently e' z — f . 5 3 — 100.
Therefore, says Heron, e' ' — 4 T 9 ^ approximately, and in III. 20
he shows how this is arrived at.
Approximation to the cube^oot of a non-cube number.
'Take the nearest cube numbers to 100 both above and
below; these are 125 and 64.
Then 125-100 = 25,
and 100- 64 = 36.
Multiply 5 into 36; this gives 180. Add 100, making 280.
(Divide 180 by 280); this gives T 9 5 . Add this to the side of
the smaller cube : this gives 4 T \ . This is as nearly as possible
the cube root ("cubic side") of 100 units.'
We have to conjecture Heron's formula from this example.
Generally, if a 3 < A < (a + l) 3 , suppose that A—a 3 = d 1 , and
(a+1) 3 — A = d 2 . The best suggestion that has been made
is Wertheim's, 1 namely that Heron's formula for the approxi-
mate cube root was a+ ; — - — r—/ — — r - The 5 multiplied
(a+\)d 1 + ad 2
into the 36 might indeed have been the square root of 25 or
Vd 2 , and the 100 added to the 180 in the denominator of the
fraction might have been the original number 100 (A) and not
4 .25 or ad 2 , but Wertheim's conjecture is the more satisfactory
because it can be evolved out of quite elementary considera-
tions. This is shown by G. Enestrom as follows. 2 Using the
same notation, Enestrom further supposes that x is the exact
value of \^A } and that {x — af — 8 V \a+ 1 — xf = 8 2 .
Thus
8 l = x 3 — 3x 2 a + 3xa 2 — a [i , and 3ax(x — a) = x' 6 — a? — S 1 = d l — 8 V
Similarly from 8 2 = (a + 1 — xf we derive
3(a+l)x{a + l-x) = (a+lf-x' d -8 2 = d 2 — 8 2 .
Therefore
d 2 -8 2 __ 3(a + \)x(a+\ —x) _ (a+1) {l-(x-a)}
d 1 — 8 l 3ax{x — a) a(x — a)
a+1 a+1
a(x — a) a '
1 Zeitschr.f. Math. u. Physik, xliv, 1899, hist.-litt. Abt., pp. 1-3.
2 Bibliotheca Mathematics, viii 3 , 1907-8, pp. 412-13.
342 HERON OF ALEXANDRIA
and, solving for x — a, we obtain
" (a+l)(d l -8 l )-\-a(d 2 — 8 2 y
or
#A =a +
(a+l)^-^)
(a + 1) (d x - 8 X ) + a(d 2 -8 2 )
Since 8 V 8 2 are in any case the cubes of fractions, we may
neglect them for a first approximation, and we have
(a + l)d 1 + ad 2
C
i \
Xl \
D
/
/, - -
i \
h
1 * "^^*****
"'"
,
a
H K
z a
B
III. 22, which shows how to cut a frustum of a cone in a given
ratio by a section parallel to the bases, shall end our account
of the Metrica. I shall give the general formulae on the left
and Heron's case on the right. Let ABED be the frustum,
let the diameters of the bases be a, a, and the height h.
Complete the cone, and let the height of GDE be x.
Suppose that the frustum has to be cut by a plane FG in
such a way that
(frustum DG) : (frustum FB) — m : n.
In the case taken by Heron
a = 28, a'= 21, h = 12, m =^4, n = 1.
Draw DH perpendicular to A B.
DIVISIONS OF
Since (DG) : (FB) = m : n,
(DB):(DG) = (m + n):m.
Now
(2)5) = '^irh (a 2 + a<x' + a' 2 ),
and (2)G) =
on
m + n
(DB).
Let ?/ be the height (CM) of the
3 one CFG.
Then DH:AH=CK:KA,
or h:±(a — a') = (x + h):^a,
whence x is known.
ConeCDE= T \7ra /2 x,
Til
cone CFG=(CDE) + (DB),
m + n
cone 045= (ODE) + (2)5).
Now, says Heron,
(CAB) + (CPE) (x + h)* + x s
(CFG) y 6
[He might have said simply
(CLE) : (CFG) = x 3 : y : \]
This gives ^/ or CM,
whence LM is known.
Now AD 2 = AH 2 + DH 2
= {±(a-a')} 2 + h 2 ,
jo that AD is known.
Therefore DF = ^^ . AD is
mown.
FIGURES 343
(DG):(FB) = 4:1,
, (D5) : (DG) -5:4.
(2)5) = 5698,
(DG) = 4558§.
14-12
x + h= —j— = 48,
and a? = 48 — 12 = 36.
(cone 02)£ T ) = 4158,
(cone (7jPG) = 4158+4558f=8716f ,
(cone CAB) =4158 + 5698 = 9856.
8716f . „
?/ = - — — • (48 J + 36 3
,7 9856 + 4158 v ;
= 8716f • -WAY- = 9 7 805,
whence ^/ = 46 approximately.
Therefore LM = y — x = 10
AD 2 = (3i) 2 + 12 2
= 156 J-,
and AD = 12|,
Therefore D2^= J§ . 12|
= 10~ 5 -
344 HERON OF ALEXANDRIA
Quadratic equations solved in Heron.
We have already met with one such equation (in Metrica
III. 4), namely x 2 — 14# + 46§ = 0, the result only (x = 8|)
being given. There are others in the Geometrlca where the
process of solution is shown.
(1) Geometrica 24, 3 (Heib.). 'Given a square such that the
sum of its area and perimeter is 896 feet: to separate the area
from the perimeter ' : i.e. x 2 + 4# = 896. Heron takes half of
4 and adds its square, completing the square on the left side.
(2) Geometrica 21, 9 and 24, 46 (Heib.) give one and the same
equation, Geom. 24, 47 another like it. 'Given the sum of
the diameter, perimeter and area of a circle, to find each
of them/
The two equations are
iid 2 +%?-d= 212,
and , \%d*+z*-d = 67|.
Our usual method is to begin by dividing by 11 throughout,
so as to leave d 2 as the first term. Heron's is to multiply by
such a number as will leave a square as the first term. In this
case he multiplies by 154, giving ll 2 c£ 2 + 58 . ll<i = 212 . 154
or 67^.154 as the case may be. Completing the square,
he obtains (11 d-\- 29) 2 = 32648 + 841 or 10395 + 841. Thus
11^ + 29=^(33489) or \/(11236), that is, 183 or 106.
Thus llc£=154or77, and d = 14 or 7, as the case may be.
Indeterminate problems in the Geometrica.
Some very interesting indeterminate problems are now
included by Heiberg in the Geometrica. 1 Two of them (chap.
24, 1-2) were included in the Geeponicus in Hultsch's edition
(sections 78, 79;; the rest are new, having been found in the
Constantinople manuscript from which Schone edited the
Metrica. As, however, these problems, to whatever period
they belong, are more akin to algebra than to mensuration,
they will be more properly described in a later chapter on
Algebra.
1 Heronia Alexandrini opera, vol. iv, p. 414. 28 sq.
THE DIOPTRA 345
The Dioptra (ire pi Stompa?).
This treatise begins with a careful description of the
dioptra, an instrument which served with the ancients for
the same purpose as a theodolite with us (chaps. 1-5). The
problems . with which the treatise goes on to deal are
(a) problems of ' heights and distances ', (6) engineering pro-
blems, (c) problems of mensuration, to which is added
(chap. 34) a description of a 'hodometer', or taxameter, con-
sisting of an arrangement of toothed wheels and endless
screws on the same axes working on the teeth of the next
wheels respectively. The book ends with the problem
(chap. 37), 'With a given force to move a given weight by
means of interacting toothed wheels', which really belongs
to mechanics, and was apparently added, like some other
problems (e.g. 31, 'to measure the outflow of, i.e. the volume
of water issuing from, a spring '), in order to make the book
more comprehensive. The essential problems dealt with are
such as the following. To determine the difference of level
between two given points (6), to draw a straight line connect-
ing two points the one of which is not visible from the other
(7), to measure the least breadth of a river (9), the distance of
two inaccessible points (10), the height of an inaccessible point
(12), to determine the difference between the heights of two
inaccessible points and the position of the straight line joining
them (13), the depth of a ditch (14) ; to bore a tunnel through
a mountain going straight from one mouth to the other (15), to
sink a shaft through a mountain perpendicularly to a canal
flowing underneath (16) ; given a subterranean canal of any
form, to find on the ground above a point from which a
vertical shaft must be sunk in order to reach a given point
on the canal (for the purpose e.g. of removing an obstruction)
(20) ; to construct a harbour on the model of a given segment
of a circle, given the ends (17), to construct a vault so that it
may have a spherical surface modelled on a given segment
(18). The mensuration problems include the following: to
measure an irregular area, which is done by inscribing a
rectilineal figure and then drawing perpendiculars to the
sides at intervals to meet the contour (23), or by drawing one
straight line across the area and erecting perpendiculars from
346 HERON OF ALEXANDRIA
that to meet the contour on both sides (24) ; given that all
the boundary stones of a certain area have disappeared except
two or three, but that the plan of the area is forthcoming,
to determine the position of the lost boundary stones (25).
Chaps. 26-8 remind us of the Metrical to divide a given
area into given parts by straight lines drawn from one point
(26) ; to measure a given area without entering it, whether
because it is thickly covered with trees, obstructed by houses,
or entry is forbidden! (27) ; chaps. 28-30 = Metrica III. 7,
III. 1, and I. 7, the last of these three propositions being the
proof of the ' formula of Heron ' for the area of a triangle in
terms of the sides. Chap. 35 shows how to find the distance
between Rome and Alexandria along a great circle of the
earth by means of the observation of the same eclipse at
the two places, the analemma for Rome, and a concave hemi-
sphere constructed for Alexandria to show the position of the
sun at the time of the said eclipse. It is here mentioned that
the estimate by Eratosthenes of the earth's circumference in
his book On the Measurement of the Earth was the most
accurate that had been made up to date. 1 Some hold that
the chapter, like some others which have no particular con-
nexion with the real subject of the Dioptra (e.g. chaps. 31, 34,
37-8) were probably inserted by a later editor, ' in order to
make the treatise as complete as possible \ 2
The Mechanics.
It is evident that the Mechanics, as preserved in the Arabic,
is far from having kept its original form, especially in
Book I. It begins with an account of the arrangement of
toothed wheels designed to solve the problem of moving a
given weight by a given force ; this account is the same as
that given at the end of the Greek text of the Dioptra, and it
is clearly the same description as that which Pappus 3 found in
the work of Heron entitled BapovXicos ('weight-lifter') and
himself reproduced with a ratio of force to weight altered
from 5:1000 to 4:160 and with a ratio of 2 : 1 substituted for
5 : 1 in the diameters of successive wheels. It would appear
that the chapter from the BapovXKo? was inserted in place of
1 Heron, vol. iii, p. 302. 13-17. 2 lb , p. 302. 9.
3 Pappus, viii, p. 1060 sq.
THE MECHANICS 347
the first chapter or chapters of the real Mechanics which had
been lost. The treatise would doubtless begin with generalities
introductory to mechanics such as we find in the (much
interpolated) beginning of Pappus, Book VIII. It must then
apparently have dealt with the properties of circles, cylinders,
and spheres with reference to their importance in mechanics ;
for in Book II. 21 Heron says that the circle is of all figures
the most movable and most easily moved, the same thing
applying also to the cylinder and sphere, and he adds in
support of this a reference to a proof ' in the preceding Book '.
This reference may be to I. 21, but at the end of that chapter
he says that 'cylinders, even when heavy, if placed on the
ground so that they touch it in one line only, are easily
moved, and the same is true of spheres also, a matter which
we have already discussed ' ; the discussion may have come
earlier in the Book, in a chapter now lost.
The treatise, beginning with chap. 2 after the passage
interpolated from the BapovXitos, is curiously disconnected.
Chaps. 2-7 discuss the motion of circles or wheels, equal or
unequal, moving on different axes (e.g. interacting toothed
wheels), or fixed on the same axis, much after the fashion of
the Aristotelian Mechanical problems.
Aristotle s Wheel.
In particular (chap. 7) Heron attempts to explain the puzzle
of the ' Wheel of Aristotle ', which remained a puzzle up to quite
modern times, and gave rise to the proverb, ' rotam Aristotelis
magis torquere, quo magis torqueretur \ l ' The question is ', says
the Aristotelian problem 24, ' why does the greater circle roll an
equal distance with the lesser circle when they are placed about
the same centre, whereas, when they roll separately, as the
size of one is to the size of the other, so are the straight lines
traversed by them to one another V 2 Let AC, BD be quadrants
of circles with centre bounded by the same radii, and draw
tangents AE, BF at A and B. In the first case suppose the
circle BD to roll along BF till D takes the position H\ then
the radius ODC will be at right angles to AE, and C will be
at G, a point such that AG is equal to BH. In the second
1 See Van Capelle, Aristotelis quaestiones mechanicae, 1812, p. 263 sq.
2 Avist. Mechanica, 855 a 28.
348
HERON OF ALEXANDRIA
case suppose the circle A G to roll along AE till ODG takes
the position 0'FE\ then D will be at F where AE = BF.
And similarly if a whole revolution is performed and OB A is
again perpendicular to AE. Contrary, therefore, to the prin-
ciple that the greater circle moves quicker than the smaller on
the same axis, it would appear that the movement of the
smaller in this case is as quick as that of the greater, since
BH = AG, and BF = AE. Heron's explanation is that, e.g.
in the case where the larger circle rolls on AE, the lesser
circle maintains the same speed as the greater because it has
favo motions ; for if we regard the smaller circle as merely
fastened to the larger, and not rolling at all, its centre will
move to 0' traversing a distance 00' equal to AE and BF;
hence the greater circle will take the lesser with it over an
equal distance, the rolling of the lesser circle having no effect
upon this.
The parallelogram of velocities.
Heron next proves the parallelogram of velocities (chap. 8);
he takes the case of a rectangle, but the proof is applicable
generally.
The way it is put is this. A
point moves with uniform velocity
along a straight line AB, from A
to B, while at the same time AB
moves with uniform velocity always
parallel to itself with its extremity
A describing the straight line AG.
Suppose that, when the point arrives at B, the straight line
THE PARALLELOGRAM OF VELOCITIES 349
reaches the position CD. Let EF be any intermediate
position of AB, and G the position at the same instant
of the moving point on it. Then clearly AE :AC=EG: EF;
therefore AE:EG = AG: EF = AG: CD, and it follows that
G lies on the diagonal AD, which is therefore the actual path
of the moving point.
Chaps. 9-19 contain a digression on the construction of
plane and solid figures similar to given figures but greater or
less in a given ratio. Heron observes that the case of plane
figures involves the finding of a mean proportional between
two straight lines, and the case of solid figures the finding of
two mean proportionals ; in chap. 1 1 he gives his solution of
the latter problem, which is preserved in Pappus and Eutocius
as well, and has already been given above (vol. i, pp. 262-3).
The end of chap. 19 contains, quite inconsequently, the con-
struction of a toothed wheel to move on an endless screw,
after which chap. 20 makes a fresh start with some observa-
tions on weights in equilibrium on a horizontal plane but
tending to fall when the plane is inclined, and on the ready
mobility of objects of cylindrical form which touch the plane
in one line only.
Motion on an inclined plane.
When a weight is hanging freely by a rope over a pulley,
no force applied to the other end of the rope less than the
weight itself will keep it up, but, if the weight is placed on an
inclined plane, and both the plane and the portion of the
weight in contact with it are smooth, the case is different.
Suppose, e.g., that a weight in the form of a cylinder is placed
on an inclined plane so that the line in which they touch is
horizontal ; then the force required to be applied to a rope
parallel to the line of greatest slope in the plane in order to
keep the weight in equilibrium is less than the weight. For
the vertical plane passing through the line of contact between
the cylinder and the plane divides the cylinder into two
unequal parts, that on the downward side of the plane being
the greater, so that the cylinder will tend to roll down ; but
the force required to support the cylinder is the ' equivalent ',
not of the weight of the whole cylinder, but of the difference
350 HERON OF ALEXANDRIA
between the two portions into which the vertical plane cuts it
(chap. 23).
On the centre of gravity.
This brings Heron to the centre of gravity (chap. 24). Here
a definition by Posidonius, a Stoic, of the ' centre of gravity '
or ' centre of inclination ' is given, namely ' a point such that,
if the body is hung up at it, the body is divided into two
equal parts ' (he should obviously have said ' divided by any
vertical plane through the "point of suspension into two equal
parts'). But, Heron says, Archimedes distinguished between
the ' centre of gravity ' and the ' point of suspension ', defining
the latter as a point on the body such that, if the body is
hung up at it, all the parts of the body remain in equilibrium
and do not oscillate or incline in any direction. ' " Bodies", said
Archimedes, " may rest (without inclining one way or another)
with either a line, or only one point, in the body fixed ".' The
1 centre of inclination ', says Heron, ' is one single point in any
particular body to which all the vertical lines through the
points of suspension converge.' Comparing Simplicius's quo-
tation of a definition by Archimedes in his KevrpofiapiKa, to
the effect that the centre of gravity is a certain point in the
body such that, if the body is hung up by a string attached to
that point, it will remain in its position without inclining in
any direction, 1 we see that Heron directly used a certain
treatise of Archimedes. So evidently did Pappus, who has
a similar definition. Pappus also speaks of a body supported
at a point by a vertical stick : if, he says, the body is in
equilibrium, the line of the stick produced upwards must pass
through the centre of gravity. 2 Similarly Heron says that
the same principles apply when the body is supported as when
it is suspended. Taking up next (chaps. 25-31) the question
of ' supports ', he considers cases of a heavy beam or a wall
supported on a number of pillars, equidistant or not, even
or not even in number, and projecting or not projecting
beyond one or both of the extreme pillars, and finds how
much of the weight is supported on each pillar. He says
that Archimedes laid down the principles in his ' Book on
1 Simplicius on Be caelo, p. 543. 31-4, Heib.
2 Pappus, viii, p. 1032. 5-24.
ON THE CENTRE OF GRAVITY 351
Supports '. As, however, the principles are the same whether
the body is supported or hung up, it does not follow that
this was a different work from that known as wept {vy&v.
Chaps. 32-3, which are on the principles of the lever or of
weighing, end with an explanation amounting to the fact
that ' greater circles overpower smaller when their movement
is about the same centre', a proposition which Pappus says
that Archimedes proved in his work ire pi {vyoov. 1 In chap. 32,
too, Heron gives as his authority a proof given by Archimedes
in the same work. With I. 33 may be compared II. 7,
where Heron returns to the same subject of the greater and
lesser circles moving about the same centre and states the
fact that weights reciprocally proportional to their radii are
in equilibrium when suspended from opposite ends of the
horizontal diameters, observing that Archimedes proved the
proposition in his work ' On the equalization of inclination '
(presumably la-oppoiviai).
Book II. The five mechanical powers.
Heron deals with the wheel and axle, the lever, the pulley,
the wedge and the screw, and with combinations of these
powers. The description of the powers comes first, chaps. 1-6,
and then, after II. 7, the proposition above referred to, and the
theory of the several powers based upon it (chaps. 8-20).
Applications to specific cases follow. Thus it is shown how
to move a weight of 1000 talents by means of a force of
5 talents, first by the system of wheels described in the
BapovXKo?, next by a system of pulleys, and thirdly by a
combination of levers (chaps. 21-5). It is possible to combine
the different powers (other than the wedge) to produce the
same result (chap. 29). The wedge and screw are discussed
with reference to their angles (chaps. 30-1). and chap. 32 refers
to the effect of friction.
Mechanics in daily life; queries and answers.
After a prefatory chapter (33), a number of queries resem-
bling the Aristotelian problems are stated and answered
(chap. 34), e.g. 'Why do waggons with two wheels carry
a weight more easily than those with four wheels?', 'Why
1 Pappus, viii, p. 1068. 20-3.
352 HERON OF ALEXANDRIA
do great weights fall to the ground in a shorter time than
lighter ones V, ' Why does a stick break sooner when one
puts one's knee against it in the middle V, 'Why do people
use pincers rather than the hand to draw a tooth ? ', ' Why
is it easy to move weights which are suspended ? ', and
1 Why is it the more difficult to move such weights the farther
the hand is away from them, right up to the point of suspension
or a point. near it ? ', ' Why are great ships turned by a rudder
although it is so small 1 ?', 'Why do arrows penetrate armour
or metal plates but fail to penetrate cloth spread out ? '
Problems on the centre of gravity, &c.
II. 35, 36, 37 show how to find the centre of gravity of
a triangle, a quadrilateral and a pentagon respectively. Then,
assuming that a triangle of uniform thickness is supported by
a prop at each angle, Heron finds what weight is supported
by each prop, (a) when the props support the triangle only,
(b) when they support the triangle plus a given weight placed
at any point on it (chaps. 38, 39). Lastly, if known weights
are put on the triangle at each angle, he finds the centre of
gravity of the system (chap. 40) ; the problem is then extended
to the case of any polygon (chap. 41).
Book III deals with the practical construction of engines
for all sorts of purposes, machines employing pulleys with
one, two, or more supports for lifting weights, oil-presses, &c.
The Catoplrica.
This work need not detain us long. Several of the theoretical
propositions which it contains are the same as propositions
in the so-called Catoptrica of Euclid, which, as we have
seen, was in all probability the work of Theon of Alexandria
and therefore much later in date. In addition to theoretical
propositions, it contains problems the purpose of which is to
construct mirrors or combinations of mirrors of such shape
as will reflect objects in a particular way, e.g. to make the
right side appear as the right in the picture (instead of the
reverse), to enable a person to see his back or to appear in
the mirror head downwards, with face distorted, with three
eyes or two noses, and so forth. Concave and convex
THE CATOPTRTCA
353
cylindrical mirrors play a part in these arrangements. The
whole theory of course ultimately depends on the main pro-
positions 4 and 5 that the angles of incidence and reflection
are equal whether the mirror is plane or circular.
Herons proof of equality of angles of incidence and reflection.
Let AB be a plane mirror, C the eye, D the object seen.
The argument rests on the fact that nature ' does nothing in
vain'. Thus light travels in a straight line, that is, by the
quickest road. Therefore, even
when the ray is a line broken
at a point by reflection, it must
mark the shortest broken line
of the kind connecting the eye
and the object. Now, says
Heron, I maintain that the
shortest of the broken lines
(broken at the mirror) which
connect G and D is the line, as
CAD, the parts of which make equal angles with the mirror.
Join DA and produce it to meet in F the perpendicular from
C to AB. Let B be any point on the mirror other than A,
and join FB, BD.
Now LEAF = L BAD
= Z CAE, by hypothesis.
Therefore the triangles AEF, AEC, having two angles equal
and AE common, are equal in all respects.
Therefore CA = AF, and CA + AD = DF.
Since FE = EG, and BE is perpendicular to FC, BF = BG
Therefore GB + BD = FB + BD
> FD,
i.e. > GA +AD.
The proposition was of course known to Archimedes. We
gather from a scholium to the Pseudo- Euclidean Gatoptrica
that he proved it in a different way, namely by reductio ad
absurdum, thus : Denote the angles GAE, DAB by a, /? re-
spectively. Then, a is > = or < /?. Suppose a > /3. Then,
1623-2
a a
354 HERON OF ALEXANDRIA
reversing the ray so that the eye is at D instead of 0, and the
object at C instead of D, we must have fi > a. But (3 was
less than a, which is impossible. (Similarly it can be proved
that a is not less than ft.) Therefore a. = /?.
In the Pseudo-Euclidean Gatoptrica the proposition is
practically assumed ; for the third assumption or postulate
at the beginning states in effect that, in the above figure, if A
be the point of incidence, CE : EA = DH : HA (where DH is
perpendicular to AB). It follows instantaneously (Prop. 1)
that ACAE = LDAH.
If the mirror is the convex side of a circle, the same result
follows a fortiori. Let GA, AD meet
the arc at equal angles, and CB, BD at
unequal angles. Let AE be the tan-
gent at A, and complete the figure.
Then, says Heron, (the angles GAC,
BAD being by hypothesis equal), if we
subtract the equal angles GAE, BAF
from the equal angles GAC, BAD (both
pairs of angles being ' mixed ', be it
observed), we have Z E AC = I FAD. Therefore CA+AD
< CF+FD and a fortiori < CB + BD.
The problems solved (though the text is so corrupt in places
that little can be made of it) were such as the following:
11, To construct a right-handed mirror (i.e. a mirror which
makes the right side right and the left side left instead of
the opposite); 12, to construct the mirror called polyiheoron
('with many images'); 16, to construct a mirror inside the
window of a house, so that you can see in it (while inside
the room) everything that passes in the street; 18, to arrange
mirrors in a given place so that a person who approaches
cannot actually see either himself or any one else but can see
any image desired (a 'ghost-seer').
XIX
PAPPUS OF ALEXANDRIA
We have seen that the Golden Age of Greek geometry
ended with the time of Apollonius of Perga. But the influence
of Euclid, Archimedes and Apollonius continued, and for some
time there was a succession of quite competent mathematicians
who, although not originating anything of capital importance,
kept up the tradition. Besides those who were known for
particular investigations, e.g. of new curves or surfaces, there
were such men as Geminus who, it cannot be doubted, were
thoroughly familiar with the great classics. Geminus, as we
have seen, wrote a comprehensive work of almost encyclopaedic
character on the classification and content of mathematics,
including the history of the development of each subject.
But the beginning of the Christian era sees quite a different
state of things. Except in sphaeric and astronomy (Menelaus
and Ptolemy), production was limited to elementary text-
books of decidedly feeble quality. In the meantime it would
seem that the study of higher geometry languished or was
completely in abeyance, until Pappus arose to revive interest
in the subject. From the w&y in which he thinks it necessary
to describe the contents of the classical works belonging to
the Treasury of Analysis, for example, one would suppose
that by his time many of them were, if not lost, completely
forgotten, and that the great task which he set himself was
the re-establishment of geometry on its former high plane of
achievement. Presumably such interest as he was able to
arouse soon flickered out, but for us his work has an in-
estimable value as constituting, after the works of the great
mathematicians which have actually survived, the most im-
portant of all our sources.
A a 2
356 PAPPUS OF ALEXANDRIA
Date of Pappus.
Pappus lived at the end of the third century A.D. The
authority for this date is a marginal note in a Leyden manu-
script of chronological tables by Theon of Alexandria, where,
opposite to the name of Diocletian, a scholium says, ' In his
time Pappus wrote'. Diocletian reigned from 284 to 305,
and this must therefore be the period of Pappus's literary
activity. It is true that Suidas makes him a contemporary
of Theon of Alexandria, adding that they both lived under
Theodosius I (379-395). But Suidas was evidently not well
acquainted with the works of Pappus; though he mentions
a description of the earth by him and a commentary on four
Books of Ptolemy's Syntaxis, he has no word about his greatest
work, the Synagoge. As Theon also wrote a commentary on
Ptolemy and incorporated a great deal of the commentary of
Pappus, it is probable that Suidas had Theon's commentary
before him and from the association of the two names wrongly
inferred that they were contemporaries.
Works (commentaries) other than the Collection.
Besides the Synagoge, which is the main subject of this
chapter, Pappus wrote several commentaries, now lost except for
fragments which have survived in Greek or Arabic. One was
a commentary on the Elements of Euclid. This must presum-
3bh\y have been pretty complete, for, while Proclus (on Eucl. I)
quotes certain things from Pappus which may be assumed to
have come in the notes on Book I, fragments of his commen-
tary on Book X actually survive in the Arabic (see above,
vol. i, pp. 154-5, 209), and again Eutocius in his note on Archi-
medes, On the Sphere and Cylinder, I. 13, says that Pappus
explained in his commentary on the Elements how to inscribe
in a circle a polygon similar to a polygon inscribed in another
circle, which problem would no doubt be solved by Pappus, as
it is by a scholiast, in a note on XII. 1. Some of the references
by Proclus deserve passing mention. (1) Pappus said that
the converse of Post. 4 (equality of all right angles) is not
true, i.e. it is not true that all angles equal to a right angle are
themselves right, since the ' angle ' between the conterminous
arcs of two semicircles which are equal and have their
WORKS OTHER THAN THE COLLECTION 357
diameters at right angles and terminating at one point is
equal to, but is not, a right angle. 1 (2) Pappus said that,
in addition to the genuine axioms of Euclid, there were others
on record about unequals added to
equals and equals added to unequals. /* j
Others given by Pappus are (says /
Proclus) involved by the definitions, I j ^ ^
e.g. that 'all parts of the plane and of \ \f N^
the straight line coincide with one n ^ y \
another', that 'a point divides a line,
a line a surface, and a surface a solid', and" that 'the infinite
is (obtained) in magnitudes both by addition and diminution'. 2
(3) Pappus gave a pretty proof of Eucl. I. 5, which modern
editors have spoiled when introducing it into text-books. If
AB, AC are the equal sides in an isosceles triangle, Pappus
compares the triangles ABC and ACB (i.e. as if he were com-
paring the triangle ABC seen from the front with the same
triangle seen from the back), and shows that they satisfy the
conditions of I. 4, so that they are equal in all respects, whence
the result follows. 3
Marinus at the end of his commentary on Euclid's Data
refers to a commentary by Pappus on that book. m .
Pappus's commentary on Ptolemy's Syntdxis has already
been mentioned (p. 274); it seems to have extended to six
Books, if not to the whole of Ptolemy's work. The Flhrld
says that he also wrote a commentary on Ptolemy's Plani-
sphaermm, which was translated into Arabic by Thabit b.
Qurra. Pappus himself alludes to his own commentary on
the Analemma of Diodorus, in the course of which he used the
conchoid of Nicomedes for the purpose of trisecting an angle.
We come now to Pappus's great work.
The Synagoge or Collection.
(a) Character of the work; ivicle range.
Obviously written with the object of reviving the classical
Greek geometry, it covers practically the whole field. It is,
1 Proclus on Eucl. I, pp. 189-90. 2 lb., pp. 197. 6-198. 15.
3 lb., pp. 249. 20-250. 12.
358 PAPPUS OF ALEXANDRIA
however, a handbook or guide to Greek geometry rather than
an encyclopaedia ; it was intended, that is, to be read with the
original works (where still extant) rather than to enable them
to be dispensed with. Thus in the case of the treatises
included in the Treasury of Analysis there is a general intro-
duction, followed by a general account of the contents, w r ith
lemmas, &c, designed to facilitate the reading of the treatises
themselves. On the other hand, where the history of a subject
is given, e.g. that of the problem of the duplication of the
cube or the finding of the two mean proportionals, the various
solutions themselves are reproduced, presumably because they
were not easily accessible, but had to be collected from various
sources. Even when it is some accessible classic which is
being described, the opportunity is taken to give alternative
methods, or to make improvements in proofs, extensions, and
so on. Without pretending to great originality, the whole
work shows, on the part of the author, a thorough grasp of
all the subjects treated, independence of judgement, mastery
of technique ; the style is terse and clear ; in short, Pappus
stands out as an accomplished and versatile mathematician,
a worthy representative of the classical Greek geometry.
(j8) List of authors mentioned.
The immense range of the Collection can be gathered from
a mere enumeration of the names of the various mathematicians
quoted or referred to in the course of it. The greatest of
them, Euclid, Archimedes and Apollonius, are of course con-
tinually cited, others are mentioned for some particular
achievement, and in a few cases the mention of a name by
Pappus is the whole of the information we possess about the
person mentioned. In giving the list of the names occurring
in the book, it will, I think, be convenient and may economize
future references if I note in brackets the particular occasion
of the reference to the writers who are mentioned for one
achievement or as the authors of a particular book or investi-
gation. The list in alphabetical order is : Apollonius of Perga,
Archimedes, Aristaeus the elder (author of a treatise in five
Books on the Elements of Conies or of ' five Books on Solid
Loci connected with the conies '), Aristarchus of Samos (On the
THE COLLECTION 359
sizes and distances of the sun and moon), Autolycus (On the
moving sphere), Carpus of Antioch (who is quoted as having
said that Archimedes wrote only one mechanical book, that
on sphere-making, since he held the mechanical appliances
which made him famous to be nevertheless unworthy of
written description : Carpus himself, who was known as
mechanicus, applied geometry to other arts of this practical
kind), Charmandrus (who added three simple and obvious loci
to those which formed the beginning of the Plane Loci of
Apollonius), Conon of Samos, the friend of Archimedes (cited
as the propounder of a theorem about the spiral in a plane
which Archimedes proved : this would, however, seem to be
a mistake, as Archimedes says at the beginning of his treatise
that he sent certain theorems, without proofs, to Conon, who
would certainly have proved them had he lived), Demetrius of
Alexandria (mentioned as the author of a work called ' Linear
considerations', ypa/ifxtKal emo-Tcco-eL?, i.e. considerations on
curves, as to which nothing more is known), Dinostratus,
the brother of Menaechmus (cited, with Nicomedes, as having
used the curve of Hippias, to which they gave the name of
quadratrix, TeTpayoovigovo-a, for the squaring of the circle),
Diodorus (mentioned as the author of an Ancdemma), Erato-
sthenes (whose mean-finder, an appliance for finding two or
any number of geometric means, is described, and who is
further mentioned as the author of two Books ' On means '
and of a work entitled 'Loci wiih reference to means'),
Erycinus (from whose Paradoxa are quoted various problems
seeming at first sight to be inconsistent with Eucl. I. 21, it
being shown that straight lines can be drawn from two points
on the base of a triangle to a point within the triangle which
are together greater than the other two sides, provided that the
points in the base may be points other than the extremities),
Euclid, Geminus the mathematician (from whom is cited a
remark on Archimedes contained in his book ' On the classifica-
tion of the mathematical sciences ', see above, p. 223), Heraclitus
(from whom Pappus quotes an elegant solution of a vevcris
with reference to a square), Hermodorus (Pappus's son, to
whom he dedicated Books VII, VIII of his Collection), Heron
of Alexandria (whose mechanical works are extensively quoted
from), Hierius the philosopher (a contemporary of Pappus,
360 PAPPUS OF ALEXANDRIA
who is mentioned as having asked Pappus's opinion on the
attempted solution by ' plane ' methods of the problem of the two
means, which actually gives a method of approximating to
a solution 1 ), Hipparchus (quoted as practically adopting three
of the hypotheses of Aristarchus of Samos), Megethion (to
whom Pappus dedicated Book V of his Collection), Menelaus
of Alexandria (quoted as the author of Sphaerica and as having
applied the name 7rapd8o£os to a certain curve), Nicomachus
(on three means additional to the first three), Nicomedes,
Pandrosion (to whom Book III of the Collection is dedicated),
Pericles (editor of Euclid's Data), Philon of Byzantium (men-
tioned along with Heron), Philon of Tyana (mentioned as the
discoverer of certain complicated curves derived from the inter-
weaving of plectoid and other surfaces), Plato (with reference
to the five regular solids), Ptolemy, Theodosius (author of the
Sphaerica and On Bays and Nights).
(y) Translations and editions.
The first published edition of the Collection was the Latin
translation by Commandinus (Venice 1589, but dated at the
end * Pisauri apud Hieronymum Concordiam 1588'; reissued
with only the title-page changed ' Pisauri ... 1602 '). Up to
1876 portions only of the Greek text had appeared, namely
Books VII, VIII in Greek and German, by C. J. Gerhardt, 1871,
chaps. 33-105 of Book V, by Eisenmann, Paris 1824, chaps.
45-52 of Book IV in losephi Torelli Veronensis Geometrica,
1769, the remains of Book II, by John Wallis (in Opera
mathematica, III, Oxford 1699); in addition, the restorers
of works of Euclid and Apollonius from the indications
furnished by Pappus give extracts from the Greek text
relating to the particular works. Breton le Champ on Euclid's
Porisms, Halley in his edition of the Conies of Apollonius
(1710) and in his translation from the Arabic and restoration
respectively of the Be sectione rationis and Be, sectione spatii
of Apollonius (1706), Camerer on Apollonius's Tactiones (1795),
Simson and Horsley in their restorations of Apollonius's Plane
Loci and IncUnationes published in the years 1749 and 1770
respectively. In the years 1876-8 appeared the only com-
- 1 See vol. i, pp. 268-70.
THE COLLECTION. BOOKS I, II, HI 361
plete Greek text, with apparatus, Latin translation, com-
mentary, appendices and indices, by Friedrich Hultsch ; this
great edition is one of the first monuments of the revived
study of the history of Greek mathematics in the last half
of the nineteenth century, and has properly formed the model
for other definitive editions of the Greek text of the other
classical Greek mathematicians, e.g. the editions of Euclid,
Archimedes, Apollonius, &c, by Heiberg and others. The
Greek index in this edition of Pappus deserves special mention
because it largely serves as a dictionary of mathematical
terms used not only in Pappus but by the Greek mathe-
maticians generally.
(8) Summary of contents.
At the beginning of the work, Book I and the first 13 pro-
positions (out of 26) of Book II are missing. The first 13
propositions of Book II evidently, like the rest of the Book,
dealt with Apollonius's method of working with very large
numbers expressed in successive powers of the myriad, 10000.
This system has already been described (vol. i, pp. 40, 54-7).
The work of Apollonius seems to have contained 26 proposi-
tions (25 leading up to, and the 26th containing, the final
continued multiplication).
Book III consists of four sections. Section (1) is a sort of
history of the problem of finding two mean 'proportionals, in
continued proportion, between two given straight lines.
It begins with some general remarks about the distinction
between theorems and problems. Pappus observes that,
whereas the ancients called them all alike by one name, some
regarding them all as problems and others as theorems, a clear
distinction was drawn by those who favoured more exact
terminology. According to the latter a problem is that in
which it is proposed to do or construct something, a theorem
that in which, given certain hypotheses, we investigate that
which follows from and is necessarily implied by them.
Therefore he who propounds a theorem, no matter how he has
become aware of the fact which is a necessary consequence of
the premisses, must state, as the object of inquiry, the right
result and no other. On the other hand, he who propounds
362 PAPPUS OF ALEXANDRIA
a problem may bid us do something which is in fact im-
possible, and that without necessarily laying himself open
to blame or criticism. For it is part of the solver's duty
to determine the conditions under which the problem is
possible or impossible, and, ' if possible, when, how, and in
how many ways it is possible '. When, however, a man pro-
fesses to know mathematics and yet commits some elementary
blunder, he cannot escape censure. Pappus gives, as an
example, the case of an unnamed person ' who was thought to
be a great geometer' but who showed ignorance in that he
claimed to know how to solve the problem of the two mean
proportionals by 'plane' methods (i.e. by using the straight
line and circle only). He then reproduces the argument of
the anonymous person, for the purpose of showing that it
does not solve the problem as its author claims. We have
seen (vol. i, pp. 269-70) how the method, though not actually
solving the problem, does furnish a series of successive approxi-
mations to the real solution. Pappus adds a few simple
lemmas assumed in the exposition.
Next comes the passage 1 , already referred to, on the dis-
tinction drawn by the ancients between (1) plane problems or
problems which can be solved by means of the straight line
and circle, (2) solid problems, or those which require for their
solution one or more conic sections, (3) linear problems, or
those which necessitate recourse to higher curves still, curves
with a more complicated and indeed a forced or unnatural
origin (Pefitao-fiivrji/) such as spirals, quadratrices, cochloids
and cissoids, which have many surprising properties of their
own. The problem of the two mean proportionals, being
a solid problem, required for its solution either conies or some
equivalent, and, as conies could not be constructed by purely
geometrical means, various mechanical devices were invented
such as that of Eratosthenes (the mean-finder), those described
in the Mechanics of Philon and Heron, and that of Nicomedes
(who used the ' cochloidal ' curve). Pappus proceeds to give the
solutions of Eratosthenes, Nicomedes and Heron, and then adds
a fourth which he claims as his own, but which is practically
the same as that attributed by Eutocius to Sporus. All these
solutions have been given above (vol. i, pp. 258-64, 266-8).
1 Pappus, iii, p. 54. 7-22.
THE COLLECTION. BOOK III
363
Section (2). The thebry of means.
Next follows a section (pp. 69-105) on the theory of the
different kinds of means. The discussion takes its origin
from the statement of the ' second problem ', which was that
of 'exhibiting the three means' (i.e. the arithmetic, geometric
and harmonic) ' in a semicircle '. Pappus first gives a con-
struction by which another geometer (dXXo? tl?) claimed to
have solved this problem, but he does not seem to have under-
stood it, and returns to the same problem later (pp. 8,0-2).
In the meantime he begins with the definitions of the
three means and then shows how, given any two of three
terms a, b, c in arithmetical, geometrical or harmonical pro-
gression, the third can be found. The definition of the mean
(b) of three terms a, b, c in harmonic progression being that it
satisfies the relation a:c = a — b:b — c, Pappus gives alternative
definitions for the arithmetic and geometric means in corre-
sponding form, namely for the arithmetic mean a:a = a — b:b — c
and for the geometric a:b = a — b:b — c.
The construction for the harmonic mean is perhaps worth
giving. Let AB, BG be two given straight lines. At A draw
DAE perpendicular to A B, and make DA, AE equal. Join
DB, BE. From G draw GF at right
angles to AB meeting DB in F.
Join EF meeting AB in C. Then
BC is the required harmonic mean.
For
AB:BG=.DA:FG
= EA:FG
= AC-.CG
= (AB-BC):(BC-BG).
Similarly, by means of a like figure, we can find BG when
AB> BC are given, and AB when BC, BG are given (in
the latter case the perpendicular DE is drawn through G
instead of A).
Then follows a proposition that, if the three means and the
several extremes are represented in one set of lines, there must
be five of them at least, and, after a set of five such lines have
been found in the smallest possible integers, Pappus passes to
364
PAPPUS OF ALEXANDRIA
the problem of representing the three means with the respective
extremes by six lines drawn in a semicircle.
Given a semicircle on the diameter AC, and B any point on
the diameter, draw BD at right angles to A G. Let the tangent
H V
at D meet AG produced in G, and measure DH along the
tangent equal to DG. Join HB meeting the radius OD in K.
Let BF be perpendicular to OB.
Then, exactly as above, it is shown that OK is a harmonic
mean between OF and OD. Also BD is the geometric mean
between A B, BC, while OG (= OD) is the arithmetic mean
between A B, BG.
Therefore the six lines DO (= OC), OK, OF, AB, BG, BD
supply the three means with the respective extremes.
But Pappus seems to have failed to observe that the ' certain
other geometer ', who has the same figure excluding the dotted
lines, supplied the same in Jive lines. For he said that DF
is ' a harmonic mean '. It is in fact the harmonic mean
between A B, BG, as is easily seen thus.
Since ODB is a right-angled triangle, and BF perpendicular
to OD,
DF:BD = BD:DO,
DF.DO = ED* = AB.BC.
or
But DO =i(AB + BC);
therefore DF . (AB + BC) = 2AB. BG
Therefore AB . (DF- BG) = BG . (AB-DF),
that is, AB:BC= (AB - DF) : (DF- BG),
and DF is the harmonic mean between AB, BG
Consequently the Jive lines DO (= OC), DF, 9 AB, BG, BD
exhibit all the three means with the extremes.
THE COLLECTION. BOOK III 365
Pappus does not seem to have seen this, for he observes
that the geometer in question, though saying that DF is
a harmonic mean, does not say how it is a harmonic mean
or between what straight lines.
In the next chapters (pp. 84-104) Pappus, following Nico-
machus and others, defines seven more means, three of which
were ancient and the last four more modern, and shows how
we can form all ten means as linear functions of oc, ft, y, where
a, ft, y are in geometrical progression. The expositiop has
already been described (vol. i, pp. 86-9).
Section (3). The 'Paradoxes' of Erycinus.
The third section of Book III (pp. 104-30) contains a series
of propositions, all of the same sort, which are curious rather
than geometrically important. They appear to have been
taken direct from a collection of Paradoxes by one Erycinus. 1
The first set of these propositions (Props. 28-34) are connected
with Eucl. I. 21, which says that, if from the extremities
of the base of any triangle two straight lines be drawn meeting
at any point within the triangle, the straight lines are together
less than the two sides of the triangle other than the base,
but contain a greater angle. It is pointed out that, if the
straight lines are allowed to be drawn from points in the base
other than the extremities, their sum may be greater than the
other two sides of the triangle.
The first case taken is that of a right-angled triangle ABC
right-angled at B. Draw AD to any point D on BC. Measure
on it BE equal to AB, bisect AE
in F, and join FC. Then shall A
DF+FC be > BA + AC.
For EF+FC=AF + FC> AC.
Add BE and AB respectively,
and we have
BF+FC> BA + AC.
More elaborate propositions are next proved, such as the
following.
1 . In any triangle, except an equilateral triangle or an isosceles
1 Pappus, iii, p. 106. 5-9.
366 PAPPUS OF ALEXANDRIA
triangle with base less than one of the other sides, it is possible
to construct on the base and within the triangle two straight
lines meeting at a point, the sum of which is equal to the sum
of the other two sides of the triangle (Props. 29, 30).
2. In any triangle in which it is possible to construct two
straight lines from the base to one internal point the sum
of which is equal to the sum of the two sides of the triangle,
it is also possible to construct two other such straight lines the
sum of which is greater than that sum (Prop. 31).
3. Under the same conditions, if the base is greater than either
of the other two sides, two straight lines can be so constructed
from the base to an internal point which are respectively
greater than the other two sides of the triangle ; and the lines
may be constructed so as to be respectively equal to the two
sides, if one of those two sides is less than the other and each
of them is less than the base (Props. 32, 33).
4. The lines may be so constructed that their sum will bear to
the sum of the two sides of the triangle any ratio less than
2 : 1 (Prop. 34).
As examples of the proofs, we will take the case of the
scalene triangle, and prove the first and Part 1 of the third of
the above propositions for such a triangle.
In the triangle ABC with base BC let AB he greater
than AC.
Take D on BA such that BD = J (BA + AC).
B H L
On DA between D and A take any point E, and draw EF
parallel to BC. Let G be any point on EF; draw GH parallel
to AB and join GC.
THE COLLECTION. BOOK III 367
Now EA+AC > EF+FC
> EG + GC and > GC, a fortiori.
Produce GC to K so that GK = EA+AC, and with G as
centre and GK as radius describe a circle. This circle w T ill
meet EC and HG, because GH = EB > BD or DA+AC and
> GK, a fortiori.
Then HG + GL = BE+EA+AC=BA + AC.
To obtain two straight lines HG', G'L such that HG'+G'L
> BA + AC, we have only to choose G' so that HG', G'L
enclose the straight lines HG, GL completely.
Next suppose that, given a triangle A BC in which BC > BA
> AC, we are required to draw from two points on BC to
an internal point two straight lines greater respectively than
BA, AC.
With B as centre and BA as radius describe the arc AEF.
Take any point E on it, and any point D on BE produced
but within the triangle. Join DC, and produce it to G so
that DG = AC. Then with D as centre and DG as radius
describe a circle. This will meet both BC and BD because
BA > AC, and a fortiori DB > DG.
Then, if L be any point on BH, it is clear that BD, DL
are two straight lines satisfying the conditions.
A point L' on BH can be found such that DL' is equal
to A B by marking off DN on DB equal to A B and drawing
with D as centre and DiV as radius a circle meeting BH
in L'. Also, if DH be joined, DH = AC.
Propositions follow (35-9) having a similar relation to the
Postulate in Archimedes, On the Sphere and Cylinder, I,
about conterminous broken lines one of which wholly encloses
368 PAPPUS OF ALEXANDRIA
the other, i.e. it is shown that broken lines, consisting of
several straight lines, can be drawn with two points on the
base of a triangle or parallelogram as extremities, and of
greater total length than the remaining two sides of the
triangle or three sides of the parallelogram.
Props. 40-2 show that triangles or parallelograms can be
constructed with sides respectively greater than those of a given
triangle or parallelogram but having a less area.
Section (4). The inscribing of the five regular solids
in a sphere.
The fourth section of Book III (pp. 132-62) solves the
problems of inscribing each of the five regular solids in a
given sphere. After some preliminary lemmas (Props. 43-53),
Pappus attacks the substantive problems (Props. 54-8), using
the method of analysis followed by synthesis in the case of
each solid.
(a) In order to inscribe a regular pyramid or tetrahedron in
the sphere, he finds two circular sections equal and parallel
to one another, each of which contains one of two opposite
edges as its diameter. If d be the diameter of the sphere, the
parallel circular sections have d' as diameter, where d 2 — \d' 2 .
(b) In the case of the cube Pappus again finds two parallel
circular sections with diameter df such that d 2 = ^d' 2 ; a square
inscribed in one of these circles is one face of the cube and
the square with sides parallel to those of the first square
inscribed in the second circle is the opposite face.
(c) In the case of the octahedron the same two parallel circular
sections with diameter d' such that d 2 = fcT 2 are used; an
equilateral triangle inscribed in one circle is one face, and the
opposite face is an equilateral triangle inscribed in the other
circle but placed in exactly the opposite way.
(d) In the case of the icosahedron Pappus finds four parallel
circular sections each passing through three of the vertices of
the icosahedron ; two of these are small circles circumscribing
two opposite triangular faces respectively, and the other two
circles are between these two circles, parallel to them, and
equal to one another. The pairs of circles are determined in
THE COLLECTION. BOOKS III, IV 369
this way. If d be the diameter of the sphere, set out two
straight lines x, y such that d, x, y are in the ratio of the sides
of the regular pentagon, hexagon and decagon respectively
described in one and the same circle. The smaller pair of
circles have r as radius where v 2 = ^y 2 , and the larger pair
have r' as radius where r 2 — \x 2 .
(e) In the case of the dodecahedron the saw e four parallel
circular sections are drawn as in the case of the icosaheclrori.
Inscribed pentagons set the opposite way are inscribed in the
two smaller circles ; these pentagons form opposite faces.
Regular pentagons inscribed in the larger circles with vertices
at the proper points (and again set the opposite way) determine
ten more vertices of the inscribed dodecahedron.
The constructions are quite different from those in Euclid
XIII. 13, 15, 14, 16, 17 respectively, where the problem is first
to construct the particular regular solid and then to 'com-
prehend it in a sphere ', i. e. to determine the circumscribing
sphere in each case. I have set out Pappus's propositions in
detail elsewhere. 1
Book IV.
At the beginning of Book IV the title and preface are
missing, and the first section of the Book begins immediately
with an enunciation. The first section (pp. 176-208) contains
Propositions 1-12 which, with the exception of Props. 8-10,
seem to be isolated propositions given for their own sakes and
not connected by any general plan.
Section (1). Extension of the theorem of Pythagoras.
The first proposition is of great interest, being the generaliza-
tion of Eucl. I. 47, as Pappus himself calls it, which is by this
time pretty widely known to mathematicians. The enunciation
is as follows.
'If ABC be a triangle and on AB, AC any parallelograms
whatever be described, as ABLE, ACFG, and if DE, FG
produced meet in H and HA be joined, then the parallelo-
grams ABDE y ACFG are together equal to the parallelogram
1 Vide notes to Euclid's propositions in The Thirteen Books of Euclid's
Elements, pp. 473, 480, 477, 489-91, 501-3.
1523 2 B b
370
PAPPUS OF ALEXANDRIA
contained by BC, HA in an angle which is equal to the sum of
the angles ABC, DHA!
Produce HA to meet BC in K, draw BL, CM parallel to KH
meeting BE in L and FG in M, and join LNM.
Then BLHA is a parallelogram, and HA is equal and
parallel to BL.
Similarly HA, CM are equal and parallel ; therefore BL, CM
are equal and parallel.
Therefore BLMC is a parallelogram ; and its angle LBK is
equal to the sum of the angles ABC, DHA.
Now a ABBE — □ BLHA, in the same parallels,
= O BLNK, for the same reason.
Similarly □ ACFG = O jiOAffl' = □ TOCif.
Therefore, by addition, □ ABDE+C3 ACFG = a 5ZM7.
It has been observed (by Professor Cook Wilson *) that the
parallelograms on A B, AC need not necessarily be erected
outwards from AB, AC. If one of them, e.g. that on AC, be
drawn inwards, as in the second figure above, and Pappus's
construction be made, we have a similar result with a negative
sign, namely,
o BLMC = □ BLNK - o CM TO
Again, if both ABBE and ACFG were drawn inwards, their
sum would be equal to BLMC drawn outwards. Generally, if
the areas of the parallelograms described outwards are regarded
as of opposite sign to those of parallelograms drawn inwards,
1 Mathematical Gazette, vii, p. 107 (May 1913).
THE COLLECTION. BOOK IV 371
we may say that the algebraic sum of the three parallelograms
is equal to zero.
Though Pappus only takes one case, as was the Greek habit,
I see no reason to doubt that he was aware of the results
in the other possible cases.
Props. 2, 3 are noteworthy in that they use the method and
phraseology of Eucl. X, proving that a certain line in one
figure is the irrational called minor (see Eucl. X. 76), and
a certain line in another figure is ' the excess by which the
binomial exceeds the straight line which produces with a
rational area a medial whole ' (Eucl. X. 77). The propositions
4-7 and 11-12 are quite interesting as geometrical exercises,
bat their bearing is not obvious : Props. 4 and 12 are remark-
able in that they are cases of analysis followed by synthesis
applied to the proof of theorems. Props. 8-10 belong to the
subject of tangencies, being the sort of propositions that would
come as particular cases in a book such as that of Apollonius
On Contacts ; Prop. 8 shows that, if there are two equal
circles and a given point outside both, the diameter of the
circle passing through the point and touching both circles
is ' given ' ; the proof is in many places obscure and assumes
lemmas of the same kind as those given later a propos of
Apollonius's treatise; Prop. 10 purports to show how, given
three unequal circles touching one another two and two, to
find the diameter of the circle including them and touching
all three.
Section (2). On circles inscribed in the dpfirjXos
(' shoemakers knife ').
The next section (pp. 208-32), directed towards the demon-
stration of a theorem about the relative sizes of successive
circles inscribed in the apfi-qXos (shoemaker's knife), is ex-
tremely interesting and clever, and I wish that I had space
to reproduce it completely. The dpf3r)Xos, which we have
already met with in Archimedes's ' Book of Lemmas ', is
formed thus. BC is the diameter of a semicircle BGC and
BC is divided into two parts (in general unequal) at B;
semicircles are described on BD, DC as diameters on the same
side of BC as BGC is ; the figure included between the three
semicircles is the apftrjXos.
Bb2
372 PAPPUS OF ALEXANDRIA
There is, says Pappus, on record an ancient proposition to
the following effect. Let successive circles be inscribed in the
dpftrjXos touching the semicircles and one another as shown
in the figure on p. 376, their centres being A, P, ... . Then, if
Pi* Vv Vz ••• be the perpendiculars from the centres A, P, ...
on BG and d lf c£ 2 , d 3 ... the diameters of the corresponding
circles,
p 1 = d 1 , p 2 =2d 2 , p 3 = Bd B ....
He begins by some lemmas, the course of which I shall
reproduce as shortly as I can.
I. If (Fig. 1) two circles with centres A, C of which the
former is the greater touch externally at B, and another circle
with centre G touches the two circles at K, L respectively,
then KL produced cuts the circle BL again in D and meets
AC produced in a point E such that AB :BG = AE : EG.
This is easily proved, because the circular segments DL, LK
are similar, and CD is parallel to AG. Therefore
AB:BC = AK:GD = AE: EC.
Also KE.EL = EB 2 .
For AE:EC=AB:BC = AB:CF= (AE- AB) : (EC- CF)
= BE:EF.
Fig 1.
But AE:EC= KE : ED ; therefore KE:ED = BE: EF.
Therefore KE . EL : EL . ED = BE* : BE . EF.
And EL. ED = BE. EF; therefore KE. EL = EB 2 .
THE COLLECTION. BOOK IV
373
II. Let (Fig. 2) BC, BD, being in one straight line, be the
diameters of two semicircles BGC, BED, and let any circle as
FGH touch both semicircles, A being the centre of the circle.
Let M be the foot of the perpendicular from A on BC, r the
radius of the circle FGH. There are two cases according
as BD lies along BC or B lies between D and C\ i.e. in the
first case the two semicircles are the outer and one of the inner
semicircles of the apfi-qXos, while in the second case they are
the two inner semicircles; in the latter case the circle FGH
may either include the two semicircles or be entirely external
to them. Now, says Pappus, it is to be proved that
in case (1) BM:r = (BC+BD) : (BC-BD),
and in case (2) BM : r = (BC-BD) -.'(BC+BD).
We will confine ourselves to the first case, represented in
the figure (Fig. 2).
Draw through A the diameter HF parallel to BC. Then,
since the circles BGC, HGF touch at G, and BC, HF are
parallel diameters, GHB, GFC are both straight lines.
Let E be the point of contact of the circles FGH and BED;
then, similarly, BEF, HED are straight lines.
Let HK, FL be drawn perpendicular to BC.
By the similar triangles BGC, BKH we have
BC:BG = BH:BK, or CB . BK = GB . BH;
and by the similar triangles BLF, BED
BF-.BL = BD.BE, or DB.BL = FB.BE.
374 PAPPUS OF ALEXANDRIA
But QB.BH=FB.BE\
therefore GB.BK = DB. BL,
or BC:BD = BL:BK.
Therefore (BC + BD) : (BC-BD) = (BL + BK) : (BL-BK)
= 2BM:KL.
And KL = HF=2r;
therefore BM : r = (BG + BD) : (BC- BD) . (a)
It is next proved that BK . LG = AM 2 .
For, by similar triangles BKH, FLG,
BK:KH=FL.LG, or BK.LG=KH.FL
= AM 2 , (b)
Lastty, since BG : BD = BL : BK, from above,
BG:GD= BL:KL, or BL.GD = BG.KL
= BC.2r. (c)
Also BD:CD = BK:KL, or BK.GD= BD . KL
= BD.2r. (d)
III. We now (Fig. 3) take any two circles touching the
semicircles BGG, BED and one another. Let their centres be
A and P, H their point of contact, d, d' their diameters respec-
tively. Then, if AM, PN are drawn perpendicular to BG,
Pappus proves that
(AM+d):d = PN:d'.
Draw BF perpendicular to BG and therefore touching the
semicircles BGG, BED at B. Join AP, and produce it to
meet BF in F.
Now, by II. (a) above,
(BG + BD): (BC-BD) = BM:AH,
and for the same reason = BN : PH ;
it follows that AH:PH=BM: BN
= FA : FP.
THE COLLECTION. BOOK IV
375
Therefore (Lemma I), it' the two circles touch the semi-
circle BED in R, E respectively, FRE is a straight line and
EF.FR = FH\
But EF.FR = FB 2 ; therefore FH = FB.
If now BH meets PN in and MA produced in S, we have,
by similar triangles, FE:FB = PH:PO = AH: AS, whence
PH = PO and SA = AH, so that 0, S are the intersections
of PN, AM with the respective circles.
Join BP, and produce it to meet MA in K.
Now BM: BN=FA: FP
= AH-.PH, from above,
= AS:PO.
And BM:BN=BK:BP
= KS : PO.
Therefore KS — AS, and KA = <i, the diameter of the
circle EHG.
Lastly,
that is,
or
MK:KS = PN:PO,
(AM+d)'.%d = PN:%d',
(AM+d):d = PN:d'.
376
PAPPUS OF ALEXANDRIA
IV. We now come to the substantive theorem.
Let FGH be the circle touching all three semicircles (Fig. 4).
We have then, as in Lemma II,
BG.BK = BD.BL,
and for the same reason (regarding FGH as touching the
semicircles BGC, DUG)
BG . GL = GB . GK.
From the first relation we have
BG:BD = BL:BK,
N K D M L C
Fig. 4.
whence DG:BD = KL : BK, and inversely BD : DC=BK : KL,
while, from the second relation, BG : GD = GK : GL,
whence BD:DG= KL : GL.
Consequently BK : KL = KL : GL,
or BK . LG = KL 2 .
But we saw in Lemma II (b) that BK . LG = AM 2 .
Therefore KL = AM, or p x = d 1 .
For the second circle Lemma III gives us
(p 1 -¥d 1 ):d 1 = Vi'- d v
whence, since <p x = d^, £> 2 — • 2<i 2 .
For the third circle
{p 2 + d 2 ):d 2 = p s :d B ,
whence p. 6 = 3d. d .
And so on ad infinitum.
THE COLLECTION. BOOK IV 377
The same proposition holds when the successive circles,
instead of being placed between the large and one of the small
semicircles, come down between the two small semicircles.
Pappus next deals with special cases (1) where the two
smaller semicircles become straight lines perpendicular to the
diameter of the other semicircle at its extremities, (2) where
we replace one of the smaller semicircles by a straight line
through D at right angles to BC, and lastly (3) where instead
of the semicircle DUC we simply have the straight line DC
and make the first circle touch it and the two other semi-
circles.
Pappus's propositions of course include as particular cases
the partial propositions of the same kind included in the ' Book
of Lemmas' attributed to Archimedes (Props. 5, 6) ; cf. p. 102.
Sections (3) and (4). Methods of squaring the circle, and of
trisecting [or dividing in any ratio) any given angle.
The last sections of Book IV (pp. 234-302) are mainly
devoted to the solutions of the problems (1) of squaring or
rectifying the circle and (2) of trisecting any given angle
or dividing it into two parts in any ratio. To this end Pappus
gives a short account of certain curves which were used for
the purpose.
(a) The Archimedean spiral.
He begins with the spiral of Archimedes, proving some
of the fundamental properties. His method of finding the
area included (1) between the first turn and the initial line,
(2) between any radius vector on the first turn and the curve,
is worth giving because it differs from the method of Archi-
medes. It is the area of the whole first turn which Pappus
works out in detail. We will take the area up to the radius
vector OB, say.
With centre and radius OB draw the circle A' BCD.
Let BC be a certain fraction, say 1 /nth, of the arc BCD A',
and CD the same fraction, 00, OD meeting the spiral in F, E
respectively. Let KS, SV be the same fraction of a straight
line KB, the side of a square KNLR. Draw ST; VW parallel
to KN meeting the diagonal KL of the square in U, Q respec-
tively, and draw M U, PQ parallel to KR.
378
PAPPUS OF ALEXANDRIA
With as centre and OE, OF as radii draw arcs of circles
meeting OF, OB in H, G respectively.
For brevity we will now denote a cylinder in which r is the
radius of the base and h the height by (cyl. r, h) and the cone
with the same base and height by (cone r, h).
N T W
By the property of the spiral,
OB:BG = (arc A'DCB) : (arc CB)
= RK : KS
= NK : KM,
whence OB:OG = NK: NM.
Now
(sector OBO) : (sector OGF) = OB 2 : OG 2 = NK 2 : MN 2
= (cyl. KN, NT) : (cyl. MN, NT).
Similarly
(sector 00D) : (sector OEH) = (cyl. ST, TW) : (cyl. PT, TW),
and so on.
The sectors OBC, OCD ... form the sector OA'DB, and the
sectors OFG, OEH . . . form a figure inscribed to the spiral.
In like manner the cylinders {KN, TN), (ST, TW) ... form the
cylinder (KN, NL), while the cylinders (MN, NT), (PT, TW) ...
form a figure inscribed to the cone (KN, NL).
Consequently
(sector OA'DB) :(fig. inscr. in spiral)
= (cyl. KN, NL) : (fig. inscr. in cone KN, NL).
THE COLLECTION. BOOK IV 379
We have a similar proportion connecting a figure circum-
scribed to the spiral and a figure circumscribed to the cone.
By increasing n the inscribed and circumscribed figures can
be compressed together, and by the usual method of exhaustion
we have ultimately
(sector OA'DB) : (area of spiral) = (cyl. KN, NL) : (cone KN, NL)
= 3:1,
or (area of spiral cut off by OB) = $ (sector OA'DB).
The ratio of the sector OA'DB to the complete circle is that
of the angle which the radius vector describes in passing from
the position OA to the position OB to four right angles, that
is, by the property of the spiral, r : a, where r = OB, a = OA.
r
Therefore (area of spiral cut off by OB) = § - • irr
a
Similarly the area of the spiral cut off by any other radius
r
vector r = 4 — • 77- r' 2 .
3 a
Therefore (as Pappus proves in his next proposition) the
first area is to the second as r 3 to r' 3 .
Considering the areas cut off by the radii vectores at the
points where the revolving line has passed through angles
of ^tt, 7r, f 7r and 2 it respectively, we see that the areas are in
the ratio of (J) 3 , (J) 3 , (f ) 3 , 1 or 1, 8, 27, 64, so that the areas of
the spiral included in the four quadrants are in the ratio
of 1, 7, 19, 37 (Prop. 22).
(P) The conchoid of Nicomedes.
The conchoid of Nicomedes is next described (chaps. 26-7),
and it is shown (chaps. 28, 29) how it can be used to find two
geometric means between two straight lines, and consequently
to find a cube having a given ratio to a given cube (see vol. i,
pp. 260-2 and pp. 238-40, where I have also mentioned
Pappus's remark that the conchoid which he describes is the
first conchoid, while there also exist a second, a third and a
fourth which are of use for other theorems).
(y) The quadratrix.
The quadratrix is taken next (chaps. 30-2), with Sporus's
criticism questioning the construction as involving a petitio
380
PAPPUS OF ALEXANDRIA
principii. Its use for squaring the circle is attributed to
Dinostratus and Nicomedes. The whole substance of this
subsection is given above (vol. i, pp. 226-30).
Tivo constructions for the quadratrix by means of
' surface-loci '.
In the next chapters (chaps. 33, 34, Props. 28, 29) Pappus
gives two alternative ways of producing the quadratrix ' by
means of surface-loci ', for which he claims the merit that
they are geometrical rather than ' too mechanical ' as the
traditional method (of Hippias) was.
(1) The first method uses a cylindrical helix thus.
Let ABC be a quadrant of a circle with centre B, and
let BD be any radius. Suppose
that EF, drawn from a point E
on the radius BD perpendicular
to BG, is (for all such radii) in
a given ratio to the arc DC.
' I say ', says Pappus, ' that the
locus of E is a certain curve.'
Suppose a right cylinder
erected from the quadrant and
a cylindrical helix GGH drawn
upon its surface. Let DH be
the generator of this cylinder through D, meeting the helix
in H. Draw BL, EI at right angles to the plane of the
quadrant, and draw HIL parallel to BD.
Now, by the property of the helix, EI(=DH) is to the
arc GD in a given ratio. Also EF : (arc CD) = a given ratio.
Therefore the ratio EF : EI is given, And since EF, EI are
given in position, FI is given in position. But FI is perpen-
dicular to BG. Therefore FI is in a plane given in position,
and so therefore is /.
But i" is also on a certain surface described by the line LH ,
which moves always parallel to the plane ABC, with one
extremity L on BL and the other extremity H on the helix.
Therefore / lies on the intersection of this surface with the
plane through FI
THE COLLECTION. BOOK IV 381
Hence / lies on a certain curve. Therefore E, its projection
on the plane ABO, also lies on a curve.
In the particular case where the given ratio of EF to the
arc CD is equal to the ratio of BA to the arc CA, the locus of
E is a quadratrix.
[The surface described by the straight line LH is a plectoid.
The shape of it is perhaps best realized as a continuous spiral
staircase, i.e. a spiral staircase with infinitely small steps.
The quadratrix is thus produced as the orthogonal projection
of the curve in which the plectoid is intersected by a plane
through BC inclined at a given angle to the plane ABC. It is
not difficult to verify the result analytically.]
(2) The second method uses a right cylinder the base of which
is an Archimedean spiral.
Let ABC be a quadrant of a circle, as before, and EF, per-
pendicular at F to BC, a straight
line of such length that EF is
to the arc DC as A B is to the
arc ADC.
«
Let a point on AB move uni-
formly from A to B while, in the
same time, AB itself revolves
uniformly about B from the position BA to the position BC.
The point thus describes the spiral AGB. If the spiral cuts
BD in G,
BA:BG = (arc ADC) : (arc DC),
or BG : (arc DC) = BA : (arc ADC).
Therefore BG = EF.
Draw GK at right angles to the plane ABC and equal to BG.
Then GK, and therefore K, lies on a right cylinder with the
spiral as base.
But BK also lies on a conical surface with vertex B such that
its generators all make an angle of \tt with the plane ABC.
Consequently K lies on the intersection of two surfaces,
and therefore on a curve.
Through K draw LK1 parallel to BD, and let BL, EI be at
right angles to the plane ABC.
Then LKI, moving always parallel to the plane ABC, with
one extremity on BL and passing through K on a certain
382
PAPPUS OF ALEXANDRIA
curve, describes a certain plectoid, which therefore contains the
point /.
Also IE = EF, IF is perpendicular to BG, and hence IF, and
therefore 7, lies on a fixed plane through BG inclined to ABG
at an angle of ^w.
Therefore I, lying on the intersection of the plectoid and the
said plane, lies on a certain curve. So therefore does the
projection of I on ABG, i.e. the point E.
The locus of E is clearly the quadratrix.
[This result can also be verified analytically.]
(S) Digression: a spiral on a sphere.
Prop. 30 (chap. 35) is a digression on the subject of a certain
spiral described on a sphere, suggested by the discussion of
a spiral in a plane.
Take a hemisphere bounded by the great circle KLM,
with H as pole. Suppose that the quadrant of a great circle
HNK revolves uniformly about the radius HO so that K
describes the circle KLM and returns to its original position
at K, and suppose that a point moves uniformly at the same
time from H to K at such speed that the point arrives at K
at the same time that HK resumes its original position. The
point will thus describe a spiral on the surface of the sphere
between the points H and K as shown in the figure.
Pappus then sets himself to prove that the portion of the
surface of the sphere cut off towards the pole between the
spiral and the arc HNK is to the surface of the hemisphere in
THE COLLECTION. BOOK IV 383
a certain ratio shown in the second figure where ABC is
a quadrant of a circle equal to a great circle in the sphere,
namely the ratio of the segment ABC to the sector DABC.
Draw the tangent CF to the quadrant at C. With C as
centre and radius CA draw the circle AEF meeting CF in F.
Then the sector CAF is equal to the sector A DC (since
CA 2 = 2 AD 2 , while Z ACF = \ Z ADC).
It is required, therefore, to prove that, if S be the area cut
off by the spiral as above described,
S: (surface of hemisphere) = (segmt. ABC) : (sector CAF).
Let KL be a (small) fraction, say I /nth, of the circum-
ference of the circle KLM, and let HPL be the quadrant of the
great circle through H, L meeting the spiral in P. Then, by
the property of the spiral,
(arc HP) : (arc HL) = (arc KL) : (circumf . of KLM )
= l:n.
Let the small circle NPQ passing through P be described
about the pole H.
Next let FE be the same fraction, \/nth, of the arc FA
that KL is of the circumference of the circle KLM, and join EC
meeting the arc ABC in B. With C as centre and CB as
radius describe the arc BG meeting CF in G.
Then the arc CB is the same fraction, 1/^th, of the arc
CB A that the arc FE is of FA (for it is easily seen that
IFCE = \LBDC y while Z FCA = \LCDA). Therefore, since
(arc CBA) = (arc HPL), (arc CB) = (arc HP), and chord CB
= chord HP.
384 PAPPUS OF ALEXANDRIA
»
Now (sector HPN on sphere) : (sector HKL on sphere)
= (chord HP) 2 : (chord HL) 2
(a consequence of Archimedes, On Sphere and Cylinder, I. 42).
And HP* : HL 2 = CB 2 : CA 2
= CB 2 :CE 2 .
Therefore
(sector HPN) : (sector HKL) = (sector CBG) : (sector CEF).
Similarly, if the arc L1J be taken equal to the arc KL and
the great circle through H, II cuts the spiral in P',. and a small
circle described about H and through P / meets the arc HPL
in j) ; and if likewise the arc BB r is made equal to the arc BO,
and CB' is produced to meet AF in E' , while again a circular
arc with C as centre and CB' as radius meets CE in b,
(sector HP']) on sphere) : (sector HLU on sphere)
= (sector CB'b) : (sector (7^^).
And so on.
Ultimately then we shall get a figure consisting of sectors
on the sphere circumscribed about the area S of the spiral and
a figure consisting of sectors of circles circumscribed about the
segment CB A ; and in like .manner we shall have inscribed
figures in each case similarly made up.
The method of exhaustion will then give
$: (surface of hemisphere) = (segmt. ABC) : (sector CAF)
= (segmt. ABC) : (sector DAC).
[We may, as an illustration, give the analytical equivalent
of this proposition. If p, a> be the spherical coordinates of P
with reference to H as pole and the arc HNK as polar axis,
the equation of Pappus's curve is obviously co = 4 p.
If now the radius of the sphere is taken as unity, we have as
the element of area
dA. — dec (1 —cos/)) — 4dp(l — cos/o).
rh-
Therefore A =
idp (1 —cos/)) = 2 7T — 4.
THE COLLECTION. BOOK IV 385
Therefore
A 2tt-4 &*■ — ■ i
(surface of hemisphere) 2n \iv
(segment ABC) -,
"" (sector I) ABC) * J
The second part of the last section of Book IV (chaps. 36-41,
pp. 270-302) is mainly concerned with the problem of tri-
secting any given angle or dividing it into parts in any given
ratio. Pappus begins with another account of the distinction
between plane, solid and linear problems (cf . Book III, chaps.
20-2) according as they require for their solution (1) the
straight line and circle only, (2) conies or their equivalent,
(3) higher curves still, 'which have a more complicated and
forced (or unnatural) origin, being produced from more
irregular surfaces and involved motions. Such are the curves
which are discovered in the so-called loci on surfaces, as
well as others more complicated still and many in number
discovered by Demetrius of Alexandria in his Linear con-
siderations and by Philon of Tyana by means of the inter-
lacing of plectoids and other surfaces of all sorts, all of which
curves possess many remarkable properties peculiar to them.
Some of these curves have been thought bv the more recent
writers to be worthy of considerable discussion ; one of them is
that which also received from Menelaus the name of the
paradoxical curve. Others of the same class are spirals,
quadratrices, cochloids and cissoids.' He adds the often-quoted
reflection on the error committed by geometers when they
solve a problem by means of an ' inappropriate class ' (of
curve or its equivalent), illustrating this by the use in
Apollonius, Book V, of a rectangular hyperbola for finding the
feet of normals to a parabola passing through one point
(where a circle would serve the purpose), and by the assump-
tion by Archimedes of a solid vevcris in his book On Spirals
(see above, pp. 65-8).
Trisection (or division in any ratio) of any angle.
The method of trisecting any angle based on a certain vevo-i y
is next described, with the solution of the vevo-i$ itself by
1523 ? C C
386 PAPPUS OF ALEXANDRIA
means of a hyperbola which has to be constructed from certain
data, namely the asymptotes and a certain point through
which the curve must pass (this easy construction is given in
Prop. 33, chap. 41-2). Then the problem is directly solved
(chaps. 43, 44) by means of a hyperbola in two ways prac-
tically equivalent, the hyperbola being determined in the one
case by the ordinary Apollonian property, but in the other by
means of the focus-directrix property. Solutions follow of
the problem of dividing any angle in a given ratio by means
(1) of the quadratrix, (2) of the spiral of Archimedes (chaps.
45, 46). All these solutions have been sufficiently described
above (vol. i, pp. 235-7, 241-3, 225-7).
Some problems follow (chaps. 47-51) depending on these
results, namely those of constructing an isosceles triangle in
which either of the base angles has a given ratio to the vertical
angle (Prop. 37), inscribing in a circle a regular polygon of
any number of sides (Prop. 38), drawing a circle the circum-
ference of which shall be equal to a given straight line (Prop.
39), constructing on a given straight line AB a segment of
a circle such that the arc of the segment may have a given
ratio to the base (Prop. 40), and constructing an angle incom-
mensurable with a given angle (Prop. 41).
Section (5). Solution of the v ever is of Archimedes, * On Spirals',
Pro}). 8, by means of conies.
Book IV concludes with the solution of the vevcri? which,
according to Pappus, Archimedes unnecessarily assumed in
On Spirals, Prop. 8. Archimedes's assumption is this. Given
a circle, a chord (BC) in it less than the diameter, and a point
A on the circle the perpendicular from which to BC cuts BC
in a point D such that BD > DO and meets the circle again
in E, it is possible to draw through A a straight line ARP
cutting BC in R and the circle in P in such a way that RP
shall be equal to DE (or, in the phraseology of yeva-ei?, to
place between the straight line BC and the circumference
of the circle a straight line equal to DE and verging
towards A).
Pappus makes the problem rather more general by not
requiring PR to be equal to DE, but making it of any given
THE COLLECTION. BOOK IV
387
length (consistent with a real solution). The problem is best
exhibited by means of analytical geometry.
If BD = a, DC = b, AD = c (so that DE = ab/c), we have
to find the point R on BC such that AR produced solves the
problem by making PR equal to k, say.
Let DR = x. Then, since BR.RC = PR.RA, we have
(a-x)(b + x) = k^{c 2 + x 2 ).
An obvious expedient is to put y for V(c 2 + x 2 ), when
we have
(a — x)(b + x) = ley, { 1 )
and
' 2 = c 2 + x l
y = c* + x\ (2)
These equations represent a parabola and a hyperbola
respectively, and Pappus does in fact solve the problem by
means of the intersection of a parabola and a hyperbola ; one
of his preliminary lemmas is, however, again a little more
general. In the above figure y is represented by RQ.
The first lemma of Pappus (Prop. 42, p. 298) states that, if
from a given point A any straight line be drawn meeting
a straight line BC given in position in R, and ii'RQ be drawn
at right angles to BC and of length bearing a given ratio
to AR, the locus of Q is a hyperbola.
For c(raw AD perpendicular to BC and produce it to A'
so that
QR : RA — A'D\ DA = the given ratio,
cc 2
388 PAPPUS OF ALEXANDRIA
Measure DA" along DA equal to DA'.
Then, if QN be perpendicular to AD,
(AR 2 -AD 2 ):(QR 2 -A'D 2 ) = (const.),
that is, QN 2 : A'N . A"N = (const),
and the locus of Q is a hyperbola.
The equation of the hyperbola is clearly
x 2 = fi(y 2 — c 2 ),
where // is a constant. In the particular case taken by
Archimedes QR = RA, or fi = 1, and the hyperbola becomes
the rectangular hyperbola (2) above.
The second lemma (Prop. 43, p. 300) proves that, if BC is
given in length, and Q is such a point that, when QR is drawn
perpendicular to BC, BR . RC = k . QR, where k is a given
length, the locus of Q is a parabola.
Let be the middle point of BC, and let OK be drawn at
right angles to BC and of length such that
0C 2 = k.K0.
Let QN' be drawn perpendicular to OK.
Then QN' 2 = OR 2
= 0C 2 -BR.RC
= k . (KO - QR), by hypothesis,
= k . KN'.
Therefore the locus of Q is a parabola.
The equation of the parabola referred to DB, DE as axes of
x and y is obviously
which easily reduces to
(a — x) (b + x) = ky, as above (1).
In Archimedes's particular case k — ab/c. *
To solve the problem then we have only to draw the para-
bola and hyperbola in question, and their intersection then
gives Q, whence R, and therefore ARP, is determined.
THE COLLECTION. BOOKS IV, V 389
Book V. Preface on the Sagacity of Bees.
It is characteristic of the great Greek mathematicians that,
whenever they were free from the restraint of the technical
language of mathematics, as when for instance they had occa-
sion to write a preface, they were able to write in language of
the highest literary quality, comparable with that of the
philosophers, historians, and poets. We have only to recall
the introductions to Archimedes's treatises and the prefaces
to the different Books of Apollonius's Conies. Heron, though
severely practical, is no exception when he has any general
explanation, historical or other, to give. We have now to
note a like case in Pappus, namely the preface to Book V of
the Collection. The editor, Hultsch, draws attention to the
elegance and purity of the language and the careful writing ;
the latter is illustrated by the studied avoidance of hiatus. 1
The subject is one which a writer of taste and imagination
would naturally find attractive, namely the practical intelli-
gence shown by bees in selecting the hexagonal form for the
cells in the honeycomb. Pappus does not disappoint us ; the
passage is as attractive as the subject, and deserves to be
reproduced.
' It is of course to men that God has given the best and
most perfect notion of wisdom in general and of mathematical
science in particular, but a partial share in these things he
allotted to some of the unreasoning animals as well. To men,
as being endowed with reason, he vouchsafed that they should
do everything in the light of reason and demonstration, but to
the other animals, while denying them reason, he granted
that each of them should, by virtue of a certain natural
instinct, obtain just so much as is needful to support life.
This instinct may be observed to exist in very many other
species of living creatures, but most of all in bees. In the first
place their orderliness and their submission to the queens who
rule in their state are truly admirable, but much more admirable
still is their emulation, the cleanliness they observe in the
gathering of honey, and the forethought and housewifely care
they devote to its custody. Presumably because they know
themselves to be entrusted with the task of bringing from
the gods to the accomplished portion of mankind a share of
1 Pappus, vol. iii, p. 1233.
390 PAPPUS OF ALEXANDRIA
ambrosia in this form, they do not think it proper to pour it
carelessly on ground or wood or any other ugly and irregular
material ; but, first collecting the sweets of the most beautiful
flowers which grow on the earth, they make from them, for
the reception of the honey, the vessels which we call honey-
combs, (with cells) all equal, similar and contiguous to one
another, and hexagonal in form. And that they have con-
trived this by virtue of a certain geometrical forethought we
may infer in this way. They would necessarily think that
the figures must be such as to be contiguous to one another,
that is to say, to have their sides common, in order that no
foreign matter could enter the interstices between them and
so defile the purity of their produce. Now only three recti-
lineal figures would satisfy the condition, I mean regular
figures which are equilateral and equiangular; for the bees
would have none of the figures which are not uniform. . . .
There being then three figures capable by themselves of
exactly filling up the space about the same point, the bees by
reason of their instinctive wisdom chose for the construction
of the honeycomb the figure which has the ,most angles,
because they conceived that it would contain more honey than
either of the two others.
' Bees, then, know just this fact which is of service to them-
selves, that the hexagon is greater than the square and the
triangle and will hold more honey for the same expenditure of
material used in constructing the different figures. We, how-
ever, claiming as we do a greater share in wisdom than bees,
will investigate a problem of still wider extent, namely that,
of all equilateral and equiangular plane figures having an
equal perimeter, that which has the greater number of angles
is always greater, and the greatest plane figure of all those
which have a perimeter equal to that of the polygons is the
circle.'
Book V then is devoted to what we may call iso perimetry ',
including in the term not only the comparison of the areas of
different plane figures with the same perimeter, but that of the
contents of different solid figures with equal surfaces.
Section (1). lsoperimetry after Zenodorus,
The first section of the Book relating to plane figures
(chaps. 1-10, pp. 308-34) evidently followed very closely
the exposition of Zenodorus wepl la-ouerpcou crxv yLOLTQ&v (see
pp. 207-13, above) ; but before passing to solid figures Pappus
inserts the proposition that of all circular segments having
THE COLLECTION. BOOK V
391
the same circumference the semicircle is the greatest, with some
preliminary lemmas which deserve notice (chaps. 15, 16).
(1) ABC is a triangle right-angled at B. With C as centre
and radius CA describe the arc
AD cutting CB produced in D.
To prove that (R denoting a right
angle)
(sector CAD) : (area ABD)
> R./BCA.
Draw AF at right angles to CA meeting CD produced in F,
and draw BH perpendicular to AF. With A as centre and
A B as radius describe the arc GBE.
«
Now (area EBF) : (area EBH) > (area EBF) : (sector ABE),
and, componendo, AFBH: (EBH) > AABF: (ABE).
But (by an easy lemma which has just preceded)
AFBH: (EBH) = AABF: (ABD),
whence AABF: (ABD) > AABF: (ABE),
and (ABE) > (ABD).
Therefore (ABE) : (ABG) > (ABD) : (ABG)
> (ABD): A ABC, a fortiori.
Therefore Z BAF: Z B A C > (ABD) : A ABC,
whence, inversely, AABC:(ABD) > Z BAG: Z BAF.
and, componendo, (sector ACD) : (ABD) > R : Z BCA.
[If a. be the circular measure of /.BCA, this gives (if AC=b)
%otb 2 :(^ocb 2 — -J sin a cos a . 6 2 ) >^7r:a,
or 2a:(2a — sin2a) > tt:2oc;
that is, 0/(0 — sin 0) > n/0, where < < tt.]
(2) ABC is again a triangle right-angled at B. With C as
centre and CA as radius draw a circle AD meeting BC pro-
duced in D. To prove that
(sector CAD) : (area ABD) > R : /.ACD.
39.2
PAPPUS OF ALEXANDRIA
Draw AE at right angles to AC. With A as centre and
AC as radius describe the circle FCE meeting AB produced
in Fand AE in E.
Then, since I ACD > ICAE, (sector ACD) > (sector AGE).
Therefore (ACD) : A ABC > (ACE) : AABC
> (ACE) : (ACF), a fortiori,
> LEAC-.LCAB.
Inversely,
AABC: (ACD) < lCAB:lEAC,
and, componendo,
(ABD) : (ACD) < Z EAB : Z EAC.
Inversely, (A CD) : (ABD) > I EAC: /.EAB
>R:LACD.
We come now to the application of these lemmas to the
proposition comparing the area of a semicircle with that of
other segments of equal circumference (chaps. 17, 18).
A semicircle is the greatest of all segments of circles which
have the same circumference.
Let ABC be a semicircle with centre G, and DEF another
segment of a circle such that the circumference DEF is equal
to the circumference ABC. I say that the area of ABC is
greater than the area of DEF.
Let H be the centre of the circle DEF. Draw EHK, BG at
right angles to DF, AC respectively. Join DH, and draw
LHM parallel to DF.
THE COLLECTION. BOOK V 393
Then LH:AG = (arc LE) : (arc AB)
— (arc LE) : (arc DE)
= (sector LEE) : (sector DEE).
Also LH 2 :AG 2 = (sector Zi/i?) : (sector AGB).
Therefore the sector LHE is to the sector AGB in the
ratio duplicate of that which the sector LHE has to the
sector DHE.
Therefore
(sector LHE) : (sector DHE) = (sector DHE) : (sector AGB),
Now (1) in the case of the segment less than a semicircle
and (2) in the case of the segment greater than a semicircle
(sector EDH) : (EDK) > R:l DHE,
by the lemmas (1) and (2) respectively.
That is,
(sector EDH) : (EDK) > L LHE: L DHE
> (sector LHE) : (sector DHE)
*
> (sector EDH) : (sector AGB),
from above.
Therefore the half segment EDK is less than the half
semicircle AGB, whence the semicircle ABC is greater than
the segment DEF.
We have already described the content of Zenodorus's
treatise (pp. 207-13, above) to which, so far as plane figures
are concerned, Pappus added nothing except the above pro-
position relating to segments of circles.
Section (2). Comparison of volumes of solids having their
surfaces equal. Case of the sphere.
The portion of Book V dealing with solid figures begins
(p. 350. 20) with the statement that the philosophers who
considered that the creator gave the universe the form of a
sphere because that was the most beautiful of all shapes also
asserted that the sphere is the greatest of all solid figures
394 PAPPUS OF ALEXANDRIA
which have their surfaces equal ; this, however, they had not
proved, nor could it be proved without a long investigation.
Pappus himself does not attempt to prove that the sphere is
greater than all solids with the same surface, but only that
the sphere is greater than any of the five regular solids having
the same surface (chap. 19) and also greater than either a cone
or a cylinder of equal surface (chap. 20).
Section (3). Digression on the semi-regular solids
of Archimedes.
He begins (chap. 19) with an account of the thirteen semi-
regular- solids discovered by Archimedes, which are contained
by polygons all equilateral and all equiangular but not all
similar (see pp. 98-101, above), and he shows how to determine
the number of solid angles and the number of edges which
they have respectively ; he then gives them the go-by for his
present purpose because they are not completely regular ; still
less does he compare the sphere with any irregular solid
having an equal surface.
The sphere is greater than any of the regular solids which
has its surface equal to that of the sphere.
The proof that the sphere is greater than any of the regular
solids with surface equal to that of the sphere is the same as
that given by Zenodorus. Let P be any one of the regular solids,
S the sphere with surface equal to that of P. To prove that
S>P. Inscribe in the solid a sphere s, and suppose that r is its
radius. Then the surface of P is greater than the surface of s,
and accordingly, if R is the radius of S, R > r. But the
volume of S is equal to the cone with base equal to the surface
of S, and therefore of P, and height equal to R ; and the volume
of P is equal to the cone with base equal to the surface of P
and height equal to r. Therefore, since R>r, volume of $ >
volume of P.
Section (4). Propositions on the lines of Archimedes,
' On the Sphere and Cylinder '.
For the fact that the volume of a sphere is equal to the cone
with base equal to the surface, and height equal to the radius,
THE COLLECTION. BOOK V 395
of the sphere, Pappus quotes Archimedes, On the Sphere and
Cylinder, but thinks proper to add a series of propositions
(chaps. 20-43, pp. 362-410) on much the same lines as those of
Archimedes and leading to the same results as Archimedes
obtains for the surface of a segment of a sphere and of the whole
sphere (Prop. 28), and for the volume of a sphere (Prop. 35).
Prop. 36 (chap. 42) shows how to divide a sphere into two
segments such that their surfaces are in a given ratio and
Prop. 37 (chap. 43) proves that the volume as well as the
surface of the cylinder circumscribing a sphere is lj times
that of the sphere itself.
Among the lemmatic propositions in this section of the
Book Props. 21, 22 may be mentioned. Prop. 21 proves that,
if C, E be two points on the tangent at if to a semicircle such
that CH = HE, and if CD, EF be drawn perpendicular to the
diameter AB, then (CD + EF)CE = AB .DF; Prop. 22 proves
a like result where C, E are points on the semicircle, CD, EF
are as before perpendicular to AB, and EH is the chord of
the circle subtending the arc which with CE makes up a semi-
circle ; in this case (CD + EF)CE = EH . DF. Both results
are easily seen to be the equivalent of the trigonometrical
formula
sin (x + y) + sin (x — y) = 2 sin x cos y,
or, if certain different angles be taken as x, y,
sin # + sin?/ . , .
= cot 4(03 — y).
cos y — cos x
Section (5). Of regular solids with surfaces equal, that is
greater which has more faces.
Returning to the main problem of the Book, Pappus shows
that, of the five regular solid figures assumed to have their
surfaces equal, that is greater which has the more faces, so
that the pyramid, the cube, the octahedron, the dodecahedron
and the icosahedron of equal surface are, as regards solid
content, in ascending order of magnitude (Props. 38-56).
Pappus indicates (p. 410. 27) that 'some of the ancients' had
worked out the proofs of these propositions by the analytical
method; for himself, he will give a method of his own by
396 PAPPUS OF ALEXANDRIA
synthetical deduction, for which he claims that it is clearer
and shorter. We have first propositions (with auxiliary
lemmas) about the perpendiculars from the centre of the
circumscribing sphere to a face of (a) the octahedron, (b) the
icosahedron (Props. 39, 43), then the proposition that, if a
dodecahedron and an icosahedron be inscribed in the same
sphere, the same small circle in the sphere circumscribes both
the pentagon of the dodecahedron and the triangle of the
icosahedron (Prop. 48) ; this last is the proposition proved by
Hypsicles in the so-called ' Book XIV of Euclid ', Prop. 2, and
Pappus gives two methods of proof, the second of which (chap.
56) corresponds to that of Hypsicles. Prop. 49 proves that
twelve of the regular pentagons inscribed in a circle are together
greater than twenty of the equilateral triangles inscribed in
the same circle. The final propositions proving that the cube
is greater than the pyramid with the same surface, the octa-
hedron greater than the cube, and so on, are Props. 52-6
(chaps. 60-4), Of Pappus's auxiliary propositions, Prop. 41
is practically contained in Hypsicles's Prop. 1, and Prop. 44
in Hypsicles's last lemma; but otherwise the exposition is
different.
Book VI.
On the contents of Book VI we can be brief. It is mainly
astronomical, dealing with the treatises included in the so-
called Little Astronomy, that is, the smaller astronomical
treatises which were studied as an introduction to the great
Syntaxia of Ptolemy. The preface says that many of those
who taught the Treasury of Astronomy, through a careless
understanding of the propositions, added some things as being
necessary and omitted others as unnecessary. Pappus mentions
at this point an incorrect addition to Theodosius, Sphaerica,
III. 6, an omission from Euclid's Phaenomena, Prop. 2, an
inaccurate representation of Theodosius, On Days and Nights,
Prop. 4, and the omission later of certain other things as
being unnecessary. His object is to put these mistakes
right. Allusions are also found in the Book to Menelaus's
Sphaerica, e.g. the statement (p. 476. 16) that Menelaus in
his Sphaerica called a spherical triangle TpLnXtvpov, three-side.
THE COLLECTION. BOOKS V, VI 397
The Sphaerica of Theodosius is dealt with at some length
(chaps. 1-26, Props. 1-27), and so are the theorems of
Autolycus On the moving Sphere (chaps. 27-9), Theodosius
On Days and Nights (chaps. 30-6, Props. 29-38), Aristarchus
On the sizes and distances of the Sun and Moon (chaps. 37-40,
including a proposition, Prop. 39 with two lemmas, which is
corrupt at the end and is not really proved), Euclid's Optics
(chaps. 41-52, Props. 4 2-54), and Euclid's Phaenomena (chaps.
53-60, Props. 55-61).
Problem arising out of Euclid's 'Optics'.
There is little in the Book of general mathematical interest
except the following propositions which occur in the section on
Euclid's Optics.
Two propositions are fundamental in solid geometry,
namely :
(a) If from a point A above a plane AB be drawn perpen-
dicular to the plane, and if from B a straight line BD be
drawn perpendicular to any straight line EF in the plane,
then will AD also be perpendicular to EF (Prop. 43).
(b) If from a point A above a plane A B be drawn to the plane
but not at right angles to it, and A M be drawn perpendicular
to the plane (i.e. if BM be the orthogonal projection of BA on
the plane), the angle ABM is the least of all the angles which
AB makes with any straight lines through B, as BP, in the
plane ; the angle ABP increases as BP moves away from BM
on either side ; and, given any straight line BP making
a certain angle with BA, only one other straight line in the
plane will make the same angle with BA, namely a straight
line BP / on the other side of BM making the same angle with
it that BP does (Prop. 44).
These are the first of a series of lemmas leading up to the
main problem, the investigation of the apparent form of
a circle as seen from a point outside its plane. In Prop. 50
(= Euclid, Optics, 34) Pappus proves the fact that all the
diameters of the circle will appear equal if the straight line
drawn from the point representing the eye to the centre of
the circle is either (a) at right angles to the plane- of the circle
or (b), if not at right angles to the plane of the circle, is equal
398
PAPPUS OF ALEXANDRIA
in length to the radius of the circle. In all other cases
(Prop. 51 = Eucl. Optics, 35) the diameters will appear unequal.
Pappus's other propositions carry farther Euclid's remark
that the circle seen under these conditions will appear
deformed or distorted (Trapecnracrfiii'os), proving (Prop. 53,
pp. 588-92) that the apparent form will be an ellipse with its
centre not, ' as some think ', at the centre of the circle but
at another point in it, determined in this way. Given a circle
ABDE with centre 0, let the eye be at a point F above the
plane of the circle such that FO is neither perpendicular
to that plane nor equal to the radius of the circle. Draw FG
perpendicular to the plane of the circle and let ADG be the
diameter through G. Join AF, DF, and bisect the angle AFD
by the straight line FG meeting AD in C. Through G draw
BE perpendicular to AD, and let the tangents at B, E meet
AG produced in K. Then Pappus proves that G (not 0) is the
centre of the apparent ellipse, that AD, BE are its major and
minor axes respectively, that the ordinates to AD are parallel
to BE both really and apparently, and that the ordinates to
BE will pass through K but will appear to be parallel to AD.
Thus in the figure, G being the centre of the apparent ellipse,
it is proved that, if LGM is any straight line through G, LG is
apparently equal to CM (it is practically assumed — a proposi-
tion proved later in Book VII, Prop. 156 — that, if LK meet
the circle again in P, and if PM be drawn perpendicular to
AD to meet the circle again in M, LM passes through G).
THE COLLECTION. BOOKS VI, VII 399
The test of apparent equality is of course that the two straight
lines should subtend equal angles at F.
The main points in the proof are these. The plane through
CF, CK is perpendicular to the planes BFE, PFM and LFR ;
hence CF is perpendicular to BE, QF to PM and HF to LR,
whence BC and CE subtend equal angles at F : so do LH, HR,
and PQ, QM.
Since FC bisects the angle AFD t and AC:CD = AK:KD
(by the polar property), Z CFK is a right angle. And CF is
the intersection of two planes at right angles, namely AFK
and BFE, in the former of which FK lies; therefore KF is
perpendicular to the plane BFE, and therefore to FN. Since
therefore (by the polar property) LN : NP = IjK : KP, it
follows that the angle LFP is bisected by FN] hence LN, NP
are apparently equal.
Again LC:CM = LN:NP = LF: FP = LF: FM.
Therefore the angles LFC, CFM are equal, and LC, CM
are apparently equal.
Lastly LR:PM=LK:KP=LN:NP=LF:FP; therefore
the isosceles triangles FLR, FPM are equiangular; there-
fore the angles PFM, LFR, and consequently PFQ, LFH, are
equal. Hence LP, RM will appear to be parallel to AD.
We have, based on this proposition, an easy method of
solving Pappus's final problem (Prop. 54). ' Given a circle
ABBE and any point within it, to find outside the plane of
the circle a point from which the circle will have the appear-
ance of an ellipse with centre C'
We have only to produce the diameter AD through C to the
pole K of the chord BE perpendicular to AD and then, in
the plane through AK perpendicular to the plane of the circle,
to describe a semicircle on CK as diameter. Any point F on
this semicircle satisfies the condition.
Book VII. On the 'Treasury of Analysis'.
Book VII is of much greater importance, since it gives an
account of the books forming what was called the Treasury of
Analysis (dvaXvouevo? tottos) and, as regards those of the books
which are now lost, Pappus's account, with the hints derivable
from the large collection of lemmas supplied by him to each
400 PAPPUS OF ALEXANDRIA
book, practically constitutes our only source of information.
The Book begins (p. 634) with a definition of analysis and
synthesis which, as being the most elaborate Greek utterance
on the subject, deserves to be quoted in full.
' The so-called 'AvaXvo/xevos is, to put it shortly, a special
body of doctrine provided for the use of those who, after
finishing the ordinary Elements, are desirous of acquiring the
power of solving problems which may be set them involving
(the construction of) lines, and it is useful for this alone. It is
the work of three men, Euclid the author of the Elements,
Apollonius of Perga and Aristaeus the elder, and proceeds by
way of analysis and synthesis.'
Definition of Analysis and Synthesis.
' Analysis, then, takes that which is sought as if it were
admitted and passes from it through its successive conse-
quences to something which is admitted as the result of
synthesis : for in analysis we assume that which is sought
as if it were already done (yeyovos), and we inquire what it is
from which this results, and again what is the antecedent
cause of the latter, and so on, until by so retracing our steps
we come upon something already known or belonging to the
class of first principles, and such a method we call analysis
as being solution backwards {avaizaXiv Xvcriv).
1 But in synthesis, reversing the process, we take as already
done that which was last arrived at in the analysis and, by
arranging in their natural order as consequences what before
were antecedents, and successively connecting them one with
another, we arrive finally at the construction of what was
sought ; and this we call synthesis.
' Now analysis is of two kinds, the one directed to searching
for the truth and called theoretical, the other directed to
finding what we are told to find and called 'problematical.
(1) In the theoretical kind we assume what is sought as if
it were existent and true, after which we pass through its
successive consequences, as if they too were true and established
by virtue of our hypothesis, to something admitted : then
(a), if that something admitted is true, that which is sought
will also be true and the proof will correspond in the reverse
order to the analysis, but (6), if we come upon something
admittedly false, that which is sought will also be false.
(2) In the problematical kind we assume that which is pro-
pounded as if it were known, after which we pass through its
THE COLLECTION. BOOK VII 401
successive consequences, taking them as true, up to something
admitted : if then (a) what is admitted is possible and obtain-
able, that is, what mathematicians call given, what was
originally proposed will also be possible, and the proof will
again correspond in the reverse order to the analysis, but if (b)
we come upon something admittedly impossible, the problem
will also be impossible.'
This statement could hardly be improved upon except that
it ought to be added that each step in the chain of inference
in the analysis must be unconditionally convertible ; that is,
when in the analysis we say that, if A is true, B is true,
we must be sure that each statement is a necessary conse-
quence of the other, so that the truth of A equally follows
from the truth of B. This, however, is almost implied by
Pappus when he says that we inquire, not what it is (namely
B) which follows from A, but what it is (B) from which A
follows, and so on.
List of works in the ' Treasury of Analysis \
Pappus adds a list, in order, of the books forming the
'Ava\v6fiei>o$, namely :
' Euclid's Data, one Book, Apollonius's Cutting-off of a ratio,
two Books, Cutting-off of an area, two Books, Determinate
Section, two Books, Contacts, two Books, Euclid's Porisms,
three Books, Apollonius's Inclinations or Vergings (vtvoei?),
two Books, the same author's Plane Loci, two Books, and
Conies, eight Books, Aristaeus's Solid Loci, five Books, Euclid's
Surface-Loci, two Books, Eratosthenes's On means, two Books.
There are in all thirty-three Books, the contents of which up
to the Conies of Apollonius I have set out for your considera-
tion, including not only the number of the propositions, the
diorismi and the cases dealt with in each Book, but also the
lemmas which are required; indeed I have not, to the best
of my belief, omitted any question arising in the study of the
Books in question.'
Description of the treatises.
Then follows the short description of the contents of the
various Books down to Apollonius's Conies; no account is
given of Aristaeus's Solid Loci, Euclid's Surface-Loci and
1688.2 X> d
402 PAPPUS OF ALEXANDRIA
Eratosthenes's On means, nor are there any lemmas to these
works except two on the Surface-Loci at the end of the Book.
The contents of the various works, including those of the
lost treatises so far as they can be gathered from Pappus,
have been described in the chapters devoted to their authors,
and need not be further referred to here, except for an
addendum to the account of Apollonius's Conies which is
remarkable. Pappus has been speaking of the ' locus with
respect to three or four lines' (which is a conic), and proceeds
to say (p. 678. 26) that we may in like manner have loci with
reference to five or six or even more lines ; these had not up
to his time become generally known, though the synthesis
of one of them, not by any means the most obvious, had been
worked out and its utility shown. Suppose that there are
five or six lines, and that p 1 ,p 2 >Pa> 2h » P5 or Pi » Pi > Pz > Pa > Ph > Pe
are the lengths of straight lines drawn from a point to meet
the five or six at given angles, then, if in the first case
PiPzPz — ^PiP5 a (where X is a constant ratio and a a given
length), and in the second case p Y P 2 Pz — ^P\P$P§i the locus
of the point is in each case a certain curve given in position.
The relation could not be expressed in the same form if
there were more lines than six, because there are only three
dimensions in geometry, although certain recent writers had
allowed themselves to speak of a rectangle multiplied by
a square or a rectangle without giving any intelligible idea of
what they meant by such a thing (is Pappus here alluding to
Heron's proof of the formula for the area of a triangle in
terms of its sides given on pp. 322-3, above ?). But the system
of compounded ratios enables it to be expressed for any
number of lines thus, ^.^§ *_» ( r -^^ ) = A. Pappus
p 2 2\ a V p n /
proceeds in language not very clear (p. 680. 30) ; but the gist
seems to be that the investigation of these curves had not
attracted men of light and leading, as, for instance, the old
geometers and the best writers. Yet there were other impor-
tant discoveries still remaining to be made. For himself, he
noticed that every one in his day was occupied with the elements,
the first principles and the natural origin of the subject-
matter of investigation ; ashamed to pursue such topics, he had
himself proved propositions of much more importance and
THE COLLECTION. BOOK VII 403
utility. In justification of this statement and ' in order that
he may not appear empty-handed when leaving the subject ',
he will present his readers with the following.
(Anticipation of Guldins Theorem,)
The enunciations are not very clearly worded, but there
is no doubt as to the sense.
' Figures generated by a complete revolution of a plane figure
about an axis are in a ratio compounded (1) of the ratio
of the areas of the figures, and (2) of the ratio of the straight
lines similarly drawn to (i.e. drawn to meet at the same angles)
the axes of rotation from the respective centres of gravity.
Figures generated by incomplete revolutions are in the rcdio
compounded (1) of the rcdio of the areas of the figures and
(2) of the ratio of the arcs described by the centres of gravity
of the respective figures, the latter rcdio being itself compounded
(a) of the ratio of the straight lines similarly drawn {from
the respective centres of gravity to the axes of rotation) and
(b) of the ratio of the angles contained (i. e. described) about
the axes of revolution by the extremities of the said straight
lines (i.e. the centres of gravity).'
Here, obviously, we have the essence of the celebrated
theorem commonly attributed to P. Guldin (1577-1643),
' quantitas rotunda in viam rotationis ducta producit Pote-
statem Rotundam uno grado altiorem Potestate sive Quantitate
Rotata *}
Pappus adds that
c these propositions, which are practically one, include any
number of theorems of all sorts about curves, surfaces, and
solids, all of which are proved at once by one demonstration,
and include propositions both old and new, and in particular
those proved in the twelfth Book of these Elements. 5
Hultsch attributes the whole passage (pp. 680. 30-682. 20)
to an interpolator, I do not know for what reason; but it
seems to me that the propositions are quite beyond what
could be expected from an interpolator, indeed I know of
no Greek mathematician from Pappus's day onward except
Pappus himself who was capable of discovering such a pro-
position.
1 Centrobaryca, Lib. ii, chap, viii, Prop. 3. Viemiae 1641.
Dd2
404 PAPPUS OF ALEXANDRIA
If the passage is genuine, it seems to indicate, what is not
elsewhere confirmed, that the Collection originally contained,
or was intended to contain, twelve Books.
Lemmas to the different treatises.
After the description of the treatises forming the Treasury
of Analysis come the collections of lemmas given by Pappus
to assist the student of each of the books (except Euclid's
Data) down to Apollonius's Conies, with two isolated lemmas
to the Surface-Loci of Euclid. It is difficult to give any
summary or any general idea of these lemmas, because they
are very numerous, extremely various, and often quite diffi-
cult, requiring first-rate ability and full command of all the
resources of pure geometry. Their number is also greatly
increased by the addition of alternative proofs, often requiring
lemmas of their own, and by the separate formulation of
particular cases where by the use of algebra and conventions
with regard to sign we can make one proposition cover all the
cases. The style is admirably terse, often so condensed as to
make the argument difficult to follow without some little
filling-out ; the hand is that of a master throughout. The
only misfortune is that, the books elucidated being lost (except
the Conies and the Cutting-off of a ratio of Apollonius), it is
difficult, often impossible, to see the connexion of the lemmas
with one another and the problems of the book to which they
relate. In the circumstances, all that I can hope to do is to
indicate the types of propositions included in the lemmas and,
by way of illustration, now and then to give a proof where it
is sufficiently out of the common.
(a) Pappus begins with Lemmas to the Sectio rationis and
Sectio spatii of Apollonius (Props. 1-21, pp. 684-704). The
first two show how to divide a straight line in a given ratio,
and how, given the first, second and fourth terms of a pro-
portion between straight lines, to find the third term. The
next section (Props. 3-12 and 16) shows how to manipulate
relations between greater and less ratios by transforming
them, e.g. componendo, convertendo, &c, in the same way
as Euclid transforms equal ratios in Book V ; Prop. 1 6 proves
that, according as a : b > or < c:d, ad > or < be. Props.
THE COLLECTION. BOOK VII 405
17-20 deal with three straight lines a, b, c in geometrical
progression, showing how to mark on a straight line containing
a, b, c as segments (including the whole among 'segments'),
lengths equal to a + c ± 2 V(ac) ; the lengths are of course equal
to a 4- c + 2 b respectively. These lemmas are preliminary to
the problem (Prop. 21), Given two straight lines AB, BC
(C lying between A and B), to find a point D on BA produced
such that BD:DA=CD: (AB + BC-2 VABTBC). This is,
of course, equivalent to the quadratic equation (a + x):x
= (a — c + x):(a + c — 2 Vac), and, after marking off AE along
AD equal to the fourth term of this proportion, Pappus solves
the equation in the usual way by application of areas.
(fi) Lemmas to the ' Determinate Section ' of Apollonius.
The next set of Lemmas (Props. 22-64, pp. 704-70) belongs
to the Determinate Section of Apollonius. As we have seen
(pp. 180-1, above), this work seems to have amounted to
a Theory of Involution. Whether the application of certain
of Pappus's lemmas corresponded to the conjecture of Zeuthen
or not, we have at all events in this set of lemmas some
remarkable applications of ' geometrical algebra '. They may
be divided into groups as follows
I. Props. 22, 25, 29
If in the figure AD. DC = BD . DE\ then
BD:DE = AB.BC:AE. EC.
A QPE B
The proofs by proportions are not difficult. Prop. 29 is an
alternative proof by means of Prop. 26 (see below). The
algebraic equivalent may be expressed thus : if ax = by, then
b (a + b)(b + x)
y " (*+y){x+y) '
II. Props. 30, 32, 34.
If in the same figure AD.DE= BD. DC, then
BD : D C = A B . B E : EC . CA .
406 PAPPUS OF ALEXANDRIA
Props. 32, 34 are alternative proofs based on other lemmas
(Props. 31, 33 respectively). The algebraic equivalent may be
stated thus : if ax = by, then - = , ~ — - •
V (x + y)(a-y)
III. Props. 35, 36.
If AB.BE = CB.BD, then AB : BE = DA . AC :CE . ED,
and CB:BD = AC .CE-.AD.DE, results equivalent to the
following : if ax = by, then
a __ (ct — y) (a-b) b_ (a — b)(b-x)
x" (b-x) (y-x) n y" (a-y)(y-x)'
IV. Props. 23, 24, 31, 57, 58.
A § C E D
1 ' i ■ 1 1
If AB = CD, and E is any point in CD,
AC.CD = AE.ED + BE.EC,
and similar formulae hold for other positions of E. If E is
between B and C, AC . CD = AE . ED-BE .EC; and if E
is on AD produced, BE . EC = AE . ED + BD . DC.
V. A small group of propositions relate to a triangle ABC
with two straight lines AD, AE drawn from the vertex A to
points on the base BC in accordance with one or other of the
conditions (a) that the angles BAC, DAE are supplementary,
(b) that the angles BAE, DAC are both right angles or, as we
may add from Book VI, Prop. 12, (c) that the angles BAD,
EAC are equal. The theorems are :
In case (a) BC.CD:BE.ED = CA 2 : AE\
(b) BC.CE:BD.DE= CA 2 : AD\
(c) DC.CE:EB.BD = AC-.AB 2 .
THE COLLECTION. BOOK VII 407
Two proofs are given of the first theorem. We will give the
first (Prop. 26) because it is a case of theoretical analysis
followed by synthesis. Describe a circle about ABD : produce
EA, CA to meet the circle again in F, G, and join BF, FG.
Substituting GC . CA for BC . CD and FE . EA for BE. ED,
we have to inquire whether GC . CA : CA 2 = FE . EA : AE 2 ,
i. e. whether t GC:CA = FE: EA,
i.e. whether GA : AC = FA : AE,
i.e. whether the triangles GAF, CAE are similar or, in other
words, whether GF is parallel to BC.
But GF is parallel to BC, because, the angles BAC, DAE
being supplementary, Z DAE = Z GAB — Z GFB, while at the
same time Z DAE = suppt. of Z FAD = Z FBD.
The synthesis is obvious.
An alternative proof (Prop. 27) dispenses with the circle,
and only requires EKH to be drawn parallel to CA to meet
AB, AD in H, K.
Similarly (Prop. 28) for case (b) it is only necessary to draw
FG through D parallel to AC meeting BA in F and AE
produced in G.
Then, I FAG, Z ADF (= ID AC) being both right angles,
FD . DG = DA 2 .
Therefore CA 2 : AD 2 = CA 2 :FD.DG = (CA : FD) . (CA : DG)
= (BC:BD).(CE:DE)
= BC.CE:BD.DE.
In case (c) a circle is circumscribed to ADE cutting AB in F
and AC in G. Then, since Z FAD = L GAE, the arcs DF, EG
are equal and therefore FG is parallel to DE. The proof is
like that of case (a).
408 PAPPUS OF ALEXANDRIA
VI. Props. 37, 38.
If AB:BC = AD 2 : DC 2 ,- whether AB be "greater or less
than AD, then
AB.BG = BD 2 .
\E in the figure is a point such that ED = CD.]
A (E) D C B
i 1 1 1
A C B P (f)
i i
The algebraical equivalent is: If - = ■ ■.,"" L t then ac — b 2 .
These lemmas are subsidiary to the next (Props. 39, 40),
being used in the first proofs of them.
Props. 39, 40 prove the following:
If AGDEB be a straight line, and if
BA . AE: BD.DE = AC 2 : CD 2 ,
then AB.BD:AE.ED = BG 2 : GE 2 ;
if, again, AG .CB.AE . EB = CD 2 :DE 2 ,
then EA .AC:CB.BE = AD 2 : DB 2 .
If AB = a, BG = b, BD = c, BE— d, the algebraic equiva-
lents are the following.
a (a-d) (a-b) 2 ac b 2
c(c-d) ' ' (b-cf ' (a-d) (c-d) " (6-d) 2 '
, .„ (a — b)b (b — c) 2 .. (a — c£)(<x — 6) (a— c) 2
and it ■: ~, =■ r rr- n > then
(a — d)d (c — df bd c 2
VII. Props. 41, 42, 43.
If AD.DC=BD.DE, suppose that in Figures (1) and (2)
0) O A $ D E B
(2) A E D C B
" : 1 1 °
(3) A E B C ,p
k = AE+GB, and in Figure (3) k = AE-BG, then
k.AD = BA.AE t k.GD = BG.GE, k.BD = AB.BC,
k.DE=AE.EC.
THE COLLECTION. BOOK VII 409
The algebraical equivalents for Figures (1) and (2) re-
spectively may be written (if a — AD, b — DC, c = BD,
d = DE) :
If ab — cd, then (a±d + c±b) a = (a + c) (a±d),
(a±d + c±b) b = (c±b) (b + d),
(a±d + c±b) c = (c + a)(c±b),
(a±d + c±b) d— (a±d)(d + b).
Figure (3) gives other varieties of sign. Troubles about
sign can be avoided by measuring all lengths in one direction
from an origin outside the line. Thus, if A = a, OB = b,
&c, the proposition may be as follows :
If (d — a) (d — c) = (6 — d) (e-d) and k = e — a + b — c,
then k(d — a) — (b — a)(e — a), k(d — c) = (b — c)(e—c),
Jc(b — d) = (b—a)(b — c) and k(e— d) = (e — a)(e — c).
VIII. Props. 45-56.
More generally, if AD . DC = BD .DE and k = AE±BC,
then, if F be any point on the line, we have, according to the
position of i^in relation to A, B, (7, D, E,
±AF. FC±EF. FB = k. DF.
Algebraically, if 0A = a, OB = b ... OF — x, the equivalent
is: If (d — a) (d — c) — (b — d) (e — d), and k = (e — a) + (b — c),
then (x — a)(x — c) + (x — e)(b — x) = k (x — d).
By making x — a, b, c, e successively in this equation, we
obtain the results of Props. 41-3 above.
IX. Props. 59-64.
In this group Props. 59, 60, 63 are lemmas required for the
remarkable propositions (61, 62, 64) in which Pappus investi-
gates ' singular and minimum ' values of the ratio
AP.FDiBP.FC,
where (A, D), (B, C) are point-pairs on a straight line and F
is another point on the straight line. He finds, not only when
the ratio has the ' singular and minimum (or maximum) ' value,
410
PAPPUS OF ALEXANDRIA
but also what the value is, for three different positions of P in
relation to the four given points.
I will give, as an illustration, the first case, on account of its
elegance. It depends on the following Lemma. AEB being
a semicircle on A B as diameter, C, D any two points on A B,
and GE, DF being perpendicular to A B, let EF be joined and
produced, and let BG be drawn perpendicular to EG. To
prove that
CB.BD = BG 2 , (1)
AG.DB = FG 2 , (2)
AD . BG = EG 2 . (3)
Join GG, GD, FB, EB y AF.
(1) Since the angles at G, D are right, F, G, B, D are coney clic.
Similarly E, G, B, C are concyclic.
Therefore
£BGD = IBFI)
= I FAB
= Z FEB, in the same segment of the semicircle,
= Z GGB, in the same segment of the circle EGBG.
And the triangles GCB, DGB also have the angle GBG
common ; therefore they are similar, and GB : BG = BG : BD,
or GB.BD = BG 2 .
(2) We have AB . BD = BF 2 ;
therefore, by subtraction, AG . DB = BF 2 -BG 2 = FG 2 .
(3) Similarly AB . BG = BE 2 ;
therefore, by subtraction, from the same result (1),
AD.BG= BE 2 -BG 2 = EG 2 .
Thus the lemma gives an extremely elegant construction for
squares equal to each of the three rectangles.
THE COLLECTION. BOOK VII
411
Now suppose (A, D), (B, C) to be two point-pairs on a
straight line, and let P, another point on it, be determined by
the relation
AB.BD:AC.CD = BP 2 :CP 2 ;.
then, says Pappus, the ratio AP . PD : BP . PC is singular and
a minimum, and is equal to
AD 2 : ( VAC.BD- VTbTCD) 2 .
On iDas diameter draw a circle, and draw BF, CG perpen-
dicular to AD on opposite sides.
Then, by hypothesis, AB . BD : AC . CD = BP 2 : CP 2 ;
therefore BF 2 : CG 2 = BP 2 : OP 2 ,
or BF:CG = BP:CP,
whence the triangles FBP, GCP are similar and therefore
equiangular, so that FPG is a straight line.
Produce GO to meet the circle in H, join FH, and draw DK
perpendicular to FH produced. Draw the diameter FL and
join LH.
Now, by the lemma, FK 2 = AC . BD, and HK 2 = AB.CD;
therefore FH = FK - HK = V(AC . BD) - V(AB . CD).
Since, in the triangles FHL, PCG, the angles at H, C are
right and Z FLH= /.PGC, the triangles are similar, and
GP:PC = FL:FH = AD: FH
= AD: {V(AC.BD)- V (AB.CD)}.
But GP:1 J C=FP:PB;
therefore GP 2 : PC 2 = FP . PG : BP . PC
= ^4i J . Pi) : £P . PC.
412
PAPPUS OF ALEXANDRIA
Therefore
AP.PD.BP .PC = AD 2 : { V(AC . BD)- V{AB . CD)}*.
The proofs of Props. 62 and 64 are different, the former
being long and involved. The results are :
Prop. 62. If P is between C and D, and
AD.DB:AC.CB = DP 2 : PG\
then the ratio AP . PB : CP . PD is singular and a minimum
and is equal to { V(AC . BD) + V(AD . BC) } 2 : DC 2 .
Prop. 64. If P is on AD produced, and
AB.BD:AC.CD = BP 2 : CP 2 ,
then the ratio AP . Pi) : 5P . PC is singular and a maximum,
and is equal to AD 2 : { V^C. BD) + V(AB . CD)} 2 . ,
(y) Lemmas on the Nevveis of Apollonius.
After a few easy propositions (e.g. the equivalent of the
proposition that, if ax + x 2 = by + y 2 , then, according as a >
or < b, a + x > or < b + y), Pappus gives (Prop. 70) the
lemma leading to the solution of the vevais with regard to
the rhombus (see pp. 190-2, above), and after that the solu-
tion by one Heraclitus of the same problem with respect to
a square (Props. 71, 72, pp. 780-4). The problem is, Given a
square A BCD, to draw through B a straight line, meeting CD
in H and AD produced in E, such that HE is equal to a given
length.
The solution depends on a lemma to the effect that, if any
straight line BHE through B meets CD in H and AD pro-
duced in E, and if EF be drawn perpendicular to BE meeting
BC produced in F, then
CF 2 = BC 2 +HE 2 .
THE COLLECTION. BOOK VII 413
Draw EG perpendicular to BF.
Then the triangles BCH, EGF are similar and 'since
BC = EG) equal in all respects ; therefore EF = BH.
Now BF* = BE 2 + EF 2 ,
or BC . BF+ BF . FC = BH . BE+ BE . EH+EF 2 .
But, the angles HCF, HEF being right, H, C, F, E are
concyclic, and BC . BF = BH . BE.
Therefore, by subtraction,
BF.FC= BE. EH+EF 2
= BE.EH+BH 2
= BH.HE+EH 2 + BH 2
= EB.BH+EH 2
= FB.BC+EH 2 .
Taking away the common part, BC . GF, we have
CF 2 = BC 2 + EH 2 .
Now suppose that we have to draw BHE through B in
such a way that HE = k. Since BC, EH are both given, we
have only to determine a length x such that x 2 = BC 2 + Jc 2 ,
produce BC to F so that CF = x, draw a semicircle on BF as
diameter, produce AD to meet the semicircle in E, and join
BE. BE is thus the straight line required.
Prop. 73 (pp. 784-6) proves that, if D be the middle point
of BC, the base of an isosceles triangle ABC, then BC is the
shortest of all the straight lines through D terminated by
the straight lines A B, AC, and the nearer to BC is shorter than
the more remote.
There follows a considerable collection of lemmas mostly
showing the equality of certain intercepts made on straight
lines through one extremity of the diameter of one of two
semicircles having their diameters in a straight line, either
one including or partly including the other, or wholly ex-
ternal to one another, on the same or opposite sides of the
diameter.
414
PAPPUS OF ALEXANDRIA
I need only draw two figures by way of illustration.
In the first figure (Prop. 83), ABC, DEF being the semi-
circles, BEKC is any straight line through C cutting both;
FG is made equal to AD; AB is joined; GH is drawn per-
pendicular to BK produced. It is required to prove that
BE = KH. (This is obvious when from L, the centre of the
semicircle DEF, LM is drawn perpendicular to BK.) If E, K
coincide in the point M' of the semicircle so that B'CH' is
a tangent, then B'M' = M'H' (Props. 83, 84).
In the second figure (Prop. 91) D is the centre of the
semicircle ABG and is also the extremity of the diameter
of the semicircle DEF. If BEGF be any straight line through
F cutting both semicircles, BE — EG. This is clear, since DE
is perpendicular to BG.
The only problem of any difficulty in this section is Prop.
85 (p. 796). Given a semicircle ABG on the diameter AG
and a point D on the diameter, to draw a semicircle passing
through D and having its diameter along DC such that, if
CEB be drawn touching it at E and meeting the semicircle
ABC in B, BE shall be equal to AD.
THE COLLECTION. BOOK VII 415
The problem is reduced to a problem contained in Apollo-
nius's Determinate Section thus.
Suppose the problem solved by the semicircle DEF, BE
being equal to AD. Join E to the centre G of the semicircle
F c
DEF. Produce DA to If, making HA equal to AD. Let K
be the middle point of DC,
Since the triangles ABC, GEC are similar,
AG 2 :GC 2 = BE 2 : EC"
= AD 2 : EC 2 , by hypothesis,
= AD 2 :GC : -DG 2 (since DG = GE)
= AG 2 -AD 2 :DG 2
= HG.DG:DG 2
= EG : DG.
Therefore
HG:DG = AD 2 :GC 2 -DG 2
= AD 2 :2DC.GK.
Take a straight line Z such that AD 2 — L . 2 DC:
therefore HG:DG = L: GK,
or HG.GK = L. DG.
Therefore, given the two straight lines HD, DK (or the
three points H, D, K on a straight line), we have to find
a point G between D and K such that
HG.GK = L. DG,
which is the second ejritagma of the third Problem in the
Determinate Section of Apollonius, and therefore may be
taken as solved. (The problem is the equivalent of the
416 PAPPUS OF ALEXANDRIA
solution *of a certain quadratic equation.) Pappus observes
that the problem is always possible (requires no Siopco-fios),
and proves that it has only one solution.
(S) Lemmas on the treatise ' On contacts' by Apollonius.
These lemmas are all pretty obvious except two, which are
important, one belonging to Book I of the treatise, and the other
to Book II. The two lemmas in question have already been set
out a propos of the treatise of Apollonius (see pp. 1 82-5, above).
As, however, there are several cases of the first (Props. 105,
107, 108, 109), one case (Prop. 108, pp. 836-8), different from
that before given, may be put down here : Given a circle and
tivo 'points B, E %vithin it, to draw straight lines through D, E
to a point A on the circumference in such a way that, if they
meet the circle again in B, C, BO shall be parallel to BE.
We proceed by analysis. Suppose the problem solved and
DA,EA drawn ('inflected') to A in such a way that, if AD,
AE meet the circle again in B, O,
BO is parallel to BE.
Draw the tangent at B meeting
EB produced in F.
Then Z FBB = Z AOB = lAEB;
therefore A, E, B, F are coney clic,
and consequently
FB.BE=AB.BB.
But the rectangle AB . BB is given, since it depends only
on the position of B in relation to the circle, and the circle
is given.
Therefore the rectangle FB . BE is given.
And BE is given ; therefore FB is given, and therefore F.
If follows that the tangent FB is given in position, and
therefore B is given. Therefore BBA is given and conse-
quently AE also.
To solve the problem, therefore, we merely take F on EB
produced such that FB . BE = the given rectangle made by
the segments of any chord through B, draw the tangent FB,
join BB and produce it to A, and lastly draw AE through to
O; BO is then parallel to BE.
THE COLLECTION. BOOK VII 417
The other problem (Prop. 117, pp. 848-50) is, as we have
seen, equivalent to the following : Given a circle arid three
'points D, E, F in a straight line external to it, to inscribe in
the circle a triangle ABC such that its sides pass sever ally
through the three 'points D, E y F. For the solution, see
pp. 182-4, above.
(e) The Lemmas to the Plane Loci of Apollonius (Props.
119-26, pp. 852-64) are mostly propositions in geometrical
algebra worked out by the methods of Eucl., Books II and VI.
We may mention the following :
Prop. 122 is the well-known proposition that, if D be the
middle point of the side BC in a triangle ABC,
BA 2 + AC 2 = 2 (AD 2 + DC 2 ).
Props. 123 and 124 are two cases of the same proposition,
the enunciation being marked by an expression which is also
found in Euclid's Data. Let AB : BC be a given ratio, and
A DEC B
let the rectangle C A . AD be given ; then, if BE is a mean
proportional between DB, BC, ' the square on AE is greater
by the rectangle CA . AD than in the ratio of AB to BC to the
square on EC\ by which is meant that
A R
AE 2 = CA.AD+ ~ c . EC 2 ,
or (AE 2 - CA . AD) : EC 2 = AB : BC.
The algebraical equivalent may be expressed thus (if AB=a,
BC = b, AD = c, BE=x):
jo /, zt ,1 (aTx) 2 -(a — b)c a
It x = v(a — c)b, then ; _: — — = -*•
v ; (x + bf b
Prop. 125 is remarkable : If C, D be two points on a straight
line AB,
Aft /ID
AD 2 + ^ . DB 2 = AC 2 + AC.CB+ == . CD\
1823,8 E e
418 PAPPUS OF ALEXANDRIA
This is equivalent to the general relation between four
points on a straight line discovered by Simson and therefore
wrongly known as Stewart's theorem :
AD 2 . BC+BD 2 . CA + CD 2 . AB + BC.CA . AB = 0.
(Simson discovered this theorem for the more general case
where D is a point outside the line ABC)
An algebraical equivalent is the identity
(d _ a f (b-c) + (d - b) 2 (c-a) + (d - c ) 2 (a - b)
+ (b — c) (c — a) (a — b) = 0.
Pappus's proof of the last-mentioned lemma is perhaps
worth giving.
A c D B
C, D being two points on the straight line AB, take the
point F on it such that
FD:DB = AC:CB. (1)
Then FB : BD = AB : BC,
and (AB-FB) : (BC-BD) = AB.BC,
or AF:CD = AB:BC,
and therefore
AF.CD:CD 2 = AB:BC. [2)
From (1) we derive
AG
and from (2)
. DB 2 = FD. DB,
^. CD 2 = AF.CD.
We have now to prove that
AD 2 + BD.DF= AC 2 + AC.CB + AF.CD,
or AD 2 + BD.DF= CA.AB + AF.CD,
THE COLLECTION. BOOK VII 419
i.e. (if DA . AC be subtracted from each side)
that • AD.DC + FD.DB = AC.DB + AF.CD,
i.e. (if AF . CD be subtracted from each side)
that FD . DC+ FD.DB = AC. DB,
or * FD.CB = AC.DB:
which is true, since, by (1) above, FD : DB = AC : CB.
(£) Lemmas fo the ' Porisms ' of Euclid.
The 38 Lemmas to the For isms of Euclid form an important
collection which, of course, has been included in one form or
other in the ' restorations ' of the original treatise. Chasles x
in particular gives a classification of them, and we cannot
do better than use it in this place : '23 of the Lemmas relate
to rectilineal figures, 7 refer to the harmonic ratio of four
points, and 8 have reference to the circle.
' Of the 23 relating to rectilineal figures, 6 deal with the
quadrilateral cut by a transversal ; 6 with the equality of
the anharmonic ratios of two systems of four points arising
from the intersections of four straight lines issuing from
one point with two other straight lines ; 4 may be regarded as
expressing a property of the hexagon inscribed in two straight
lines ; 2 give the relation between the areas of two triangles
which have two angles equal or supplementary ; 4 others refer
to certain systems of straight lines; and the last is a case
of the problem of the Cutting-off of an area.'
The lemmas relating to the quadrilateral and the transversal
are 1, 2, 4, 5, 6 and 7 (Props. 127, 128, 130, 131, 132, 133).
Prop. 130 is a general proposition about any transversal
whatever, and is equivalent to one of the equations by which
we express the involution of six points. If A, A'; B, B' ;
C, C be the points in which the transversal meets the pairs of
1 Chasles, Les trois livres de Porismes d'Euclide, Paris, 1860, pp. 74 sq.
E e 2
420 PAPPUS OF ALEXANDRIA
opposite sides and the two diagonals respectively, Pappus's
resiilt is equivalent to
AB^B'C^ GA
TW7M ~ C 7 A' ' •
Props. 127, 128 are particular cases in which the transversal
is parallel to a side; in Prop. 131 the transversal passes
through the points of concourse of opposite sides, and the
result is equivalent to the fact that the two diagonals divide
into proportional parts the straight line joining the points of
concourse of opposite sides; Prop. 132 is the particular case
of Prop. 131 in which the line joining the points of concourse
of opposite sides is parallel to a diagonal; in Prop. 133 the
transversal passes through one only of the points of concourse
of opposite sides and is parallel to a diagonal, the result being
CA 2 = GB . GB\
Props. 129, 136, 137, 140, 142, 145 (Lemmas 3, 10, 11, 14, 16,
19) establish the equality of the anharmonic ratios which
four straight lines issuing from a point determine on two
transversals ; but both transversals are supposed to be drawn
from the same point on one of the four straight lines. Let
AB, AC, AD be cut by transversals HBGD, HEFG. It is
required to prove that
HE.FG HB.GD
EG.EF" HD.BC'
Pappus gives (Prop. 129) two methods of proof which are
practically equivalent. The following is the proof 'by com-
pound ratios '.
Draw HK parallel to AF meeting DA and AE produced
THE COLLECTION. BOOK VII 421
in K, L; and draw LAI parallel to AD meeting GE pro-
duced in M.
HE.FG HE FG LH A£ LH
n EG . EF~ EF ' EG ~ AF ' HK == EK '
In exactly the same way, if BE produced meets LM in M'
we prove that
HB . CD LE
Therefore
ED.BC EK
EE.FG EB.CD
EG.EF" ED.BC
(The proposition is proved for If BCD and any other trans-
versal not passing through E by applying our proposition
twice, as usual.)'
Props. 136, 142 are the reciprocal; Prop. 137 is a particular
case in which one of the transversals is parallel to one of the
straight lines, Prop. 140 a reciprocal of Prop. 137, Prop. 145
another case of Prop. 129.
The Lemmas 12, 13, 15, 17 (Props. 138, 139, 141, 143) are
equivalent to the property of the hexagon inscribed in two
straight lines, viz. that, if the vertices of a hexagon are
situate, three and three, on two straight lines, the points of
concourse of opposite sides are in a straight line ; in Props.
138, 141 the straight lines are parallel, in Props. 139, 143 not
parallel.
Lemmas 20, 21 (Props. 146, 147) prove that, when one angle
of one triangle is equal or supplementary to • one angle of
another triangle, the areas of the triangles are in the ratios
of the rectangles contained by the sides containing the equal
or supplementary angles.
The seven Lemmas 22, 23, 24, 25, 26, 27, 34 (Props. 148-53
and 160) are propositions relating to the segments of a straight
line on which two intermediate points are marked. Thus :
Props. 148, 150.
If C f D be two points on AB, then
(a) if 2AB.CD = CB 2 , AD Z ^AC 2 + DB 2 \
A C D B
i— 1 1 *
(b) if 2AC.BD = CD*, AB 2 = AD 2 + CB\
4?2 PAPPUS OF ALEXANDRIA
Props. 149, 151.
If AB . BC = BD 2 ,
then (AD±DC)BD = AD.DG,
(AD±DC)BC= DC 2 ,
A
C
B
A
i
1
C
— i
B
and (AD±DC)BA = AD 2 .
Props. 152, 153.
If AB:BC=AD 2 : DC 2 , then AB . BG = BD 2 .
DC B
-4 1 1
Prop. 160.
If AB : BC=AD : DC, then, if ^be the middle point of AC,
BE. ED = EC 2 ,
BD.DE= AD. DC,
EB.BD = AB. BC.
A £ D C B
1 1 1 —
The Lemmas about the circle include the harmonic proper-
ties of the pole and polar, whether the pole is external to the
circle (Prop. 154) or internal (Prop. 161). Prop. 155 is a
problem, Given a segment of a circle on A B as base, to inflect
straight lines AC, BC to the segment in a given ratio to one
another.
Prop. 156 is one which Pappus has already used earlier
in the Collection. It proves that the straight lines drawn
from the extremities of a chord (DE) to any point (F) of the
circumference divide harmonically the diameter (AB) perpen-
dicular to the chord. Or, if ED, FK be parallel chords, and
EF, DK meet in G, and EK, DF in H, then
AH:BB = AG:GB.
THE COLLECTION. BOOK VII
423
Since AB bisects BE perpendicularly, (arc AE) — (arc AD)
and Z.EFA = lAFD, or AF bisects the angle EFD.
1
JF
A [ — *"*
H/
'^ ^^^^^?
;)b ..--**
X — *" * ^^^
'k
Since the angle J.i^B is right, FB bisects AHFG, the supple-
ment of Z EFD.
Therefore (Eucl. VI. 3) GB : BLI = GF : FH = GzL : ^Itf,
and, alternately and inversely, AH : HB = AG : GB.
Prop. 157 is remarkable in that (without any mention of
a conic) it is practically identical with Apollonius's Conies
III. 45 about the foci of a central conic. Pappus's theorem
is as follows. Let A B be the diameter of a semicircle, and
G'x -A
from A, B let two straight lines AE, BD be drawn at right
angles to AB. Let any straight line DE meet the two perpen-
diculars in D, E and the semicircle in F. Further, let FG be
drawn at right angles to DE, meeting AB produced in G.
It is to be proved that
AG.GB = AE.BD
Since F, D, G, B are concyclic, Z BDG = Z BFG.
424 PAPPUS OF ALEXANDRIA
And, since AFB.EFG are both right angles, lBFG = lAFE.
But, since A, E, G, F are concyclic, LAFE — A AGE.
Therefore IBDG = I AGE;
and the right-angled triangles DBG, GAE are similar.
Therefore AG : AE = BD: GB,
or AG.GB = AE.DB.
In Apollonius G and the corresponding point G' on BA
produced which is obtained by drawing F'G' perpendicular to
ED (where DE meets the circle again in F') are the foci
of a central conic (in this case a hyperbola), and DE is any
tangent to the conic ; the rectangle AE . BD is of course equal
to the square on half the conjugate axis.
(77) The Lemmas to the Conies of Apollonius (pp. 918-1004)
do not call for any extended notice. There are a large number
of propositions in geometrical algebra of the usual kind,
relating to the segments of a straight line marked by a number
of points on it ; propositions about lines divided into propor-
tional segments and about similar figures ; two propositions
relating to the construction of a hyperbola (Props. 204, 205)
and a proposition (208) proving that two hyperbolas with the
same asymptotes do not meet one another. There are also
two propositions (221, 222) equivalent to an obvious trigono-
metrical formula. Let ABGD be a rectangle, and let any
straight line through A meet DC produced in E and BG
(produced if necessary) in F.
Then
EA . AF = ED . DC + CB . BF.
THE COLLECTION. BOOK VII 425
For EA 2 + AF 2 = ED 2 + DA* + AB 2 + BF 2
= ED 2 + BC 2 + C£ 2 + 5i^ 2 .
Also EA 2 + AF 2 = EF 2 + 2EA. AF.
Therefore
2EA.AF= EA 2 + AF 2 - EF 2
= ED 2 + BC 2 + CD 2 + BF 2 - EF 2
= (.ED 2 + CD 2 ) + (.BO 2 + BF 2 ) - EF 2
= EC 2 + 2ED.DC+CF 2 + 2CB.BF-EF 2
= 2ED.DC+2CB.BF;
i.e. EA . 4 F = ED . DC+CB . Itf 7 .
This is equivalent to sec cosec = tan 6 + cot 6.
The algebraical equivalents of some of the results obtained
by the usual geometrical algebra may be added.
Props. 178, 179, 192-4.
(a + 2b)a + {b + x) (b-x) = (a + b + x)(a + b — x).
Prop. 195. 4a 2 = 2{(a-x) (a + x) + (a-y) (a + y) + x 2 + y 2 \.
Prop. 196.
{a + b-x) 2 + (a + b + x) 2 = (x-b) 2 + (x + b) 2 +2(a + 2b)a.
Props. 197, 199, 198.
If (x + y + a)a + x 2 = (a + x) 2 , \
or if (x + y + a) a + # 2 = (a + 2/) 2 , L then x = y.
or if (a; + 2/ — a) a + (# — a) 2 = 2/ 2 , ,
2b + a b + x
b — x
a
and
Props. 200, 201. If (a + b)x = b 2 , then
(2b + a)a = (a + 6) (a + & — a?).
Prop. 207. If (a + b)b — 2a 2 , then a = b.
(6) The two Lemmas to the Surface- Loci of Euclid have
already been mentioned as significant. The first has the
appearance of being a general enunciation, such as Pappus
426 PAPPUS OF ALEXANDRIA
is fond of giving, to cover a class of propositions. The
enunciation may be translated as follows : ' If A B be a straight
line, and CD a straight line parallel to a straight line given in
position, and if the ratio AD . DB : DC 2 be given, the point C
lies on a conic section. If now AB be no longer given in
position, and the points A, B are no longer given but lie
(respectively) on straight lines AE, EB given in position, the
point G raised above (the plane containing AE, EB) lies on
a surface given in position. And this was proved.' Tannery
was the first to explain this intelligibly ;
and his interpretation only requires the
very slight change in the text of sub-
stituting evOeiais for evOeia in the phrase
yivrjTai St 777)0? decrei evdeta reus AE, EB.
It is not clear whether, when AB ceases
to be given in 'position, it is still given
in length. If it is given in length and A, B move on the lines
AE, EB respectively, the surface which is the locus of G is
a complicated one such as Euclid would hardly have been
in a position to investigate. But two possible cases are
indicated which he may have discussed, (1) that in which AB
moves always parallel to itself and varies in length accord-
ingly, (2) that in which the two lines on which A, B move are
parallel instead of meeting at a point. The loci in these two
cases would of course be a cone and a cylinder respectively.
The second Lemma is still more important, since it is the
lirst statement on record of the focus-directrix property of
the three conic sections. The proof, after Pappus, has been
set out above (pp. 119-21).
(1) An unallocated Lemma.
Book VII ends (pp. 1016-18) with a lemma which is not
given under any particular treatise belonging to the Treasury
of Analysis, but is simply called 'Lemma to the 'Ai/a\v6/jLeuos\
If ABC be a triangle right-angled at B, and AB, BG be
divided at F, G so that AF : FB = BG : GC = AB: BC, and
if AEG, CEF be joined and BE joined and produced to D,
then shall BD be perpendicular to AC.
The text is unsatisfactory, for there is a long interpolation
containing an attempt at a proof by reductio ad absurdum ;
THE COLLECTION. BOOKS VII, VIII 427
but the genuine proof is indicated, although it breaks off
before it is quite complete.
Since AF:FB = BG:GC,
AB:FB = BC:GC t
or AB : BC = FB : GO.
But, by hypothesis, AB:BC=BG:GC;
therefore BF = BG.
From this point the proof apparently proceeded by analysis.
' Suppose it done ' (y^yoverco), i.e. suppose the proposition true,
and BED perpendicular to AC.
Then, by similarity of triangles, AD : DB = AB : BC ;
therefore AF'.FB—AD.DB, and consequently the angle
ADB is bisected by DF.
Similarly the angle BDC is bisected by DG.
Therefore each of the angles BDF, BDG is half a right
angle, and consequently the angle FDG is a right angle.
Therefore B, G, D, F are concyclic ; and, since the angles
FDB, BDG are equal, FB = BG.
This is of course the result above proved.
Evidently the interpolator tried to clinch the argument by
proving that the angle BD A could not be anything but a right
angle.
Book VIII.
Book VIII of the Collection is mainly on mechanics, although
it contains, in addition, some propositions of purely geometrical
interest.
428 PAPPUS OF ALEXANDRIA
Historical 'preface.
It begins with an interesting preface on the claim of
theoretical mechanics, as distinct from the merely practical
or industrial, to be regarded as a mathematical subject.
Archimedes, Philon, Heron of Alexandria are referred to as
the principal exponents of the science, while Carpus of Antioch
is also mentioned as having applied geometry to ' certain
(practical) arts'.
The date of Carpus is uncertain, though it is probable that
he came after Geminus; the most likely date seems to be the
first or second century A. D. Simplicius gives the authority of
Iamblichus for the statement that Carpus squared the circle
by means of a certain curve, which he simply called a curve
generated by a double motion. 1 Proclus calls him ' Carpus the
writer on mechanics (o firjxaviKos) ', and quotes from a work of
his on Astronomy some remarks about the relation between
problems and theorems and the 'priority in order' of the
former. 2 Proclus also mentions him as having held that an
angle belongs to the category of quantity (ttoo-ou), since it
represents a sort of ' distance ' between the two lines forming
it, this distance being ' extended one way ' {k<f> %v SiecrTcos)
though in a different sense from that in which a line represents
extension one way, so that Carpus's view appeared to be ' the
greatest possible paradox ' 3 ; Carpus seems in reality to have
been anticipating the modern view of an angle as representing
divergence rather than distance, and to have meant by eft ef
in one sense (rotationally), as distinct from one way or in one
dimension (linearly).
Pappus tells us that Heron distinguished the logical, i.e.
theoretical, part of mechanics from the practical or manual
(xtLpovpyiKov), the former being made up of geometry, arith-
metic, astronomy and physics, the latter of work in metal,
architecture, carpentering and painting ; the man who had
been trained from his youth up in the sciences aforesaid as well
as practised in the said arts would naturally prove the best
architect and inventor of mechanical devices, but, as it is diffi-
cult or impossible for the same person to do both the necessary
1 Simplicius on Arist. Categ., p. 192, Kalbfleisch.
2 Proclus on Eucl. I, pp. 241-3. y lb., pp. 125. 25-126. 6.
THE COLLECTION. BOOK VIII 429
mathematics and the practical work, he who has not the former
must perforce use the resources which practical experience in
his particular art or craft gives him. Other varieties of
mechanical work included by the ancients under the general
term mechanics were (1) the use of the mechanical powers,
or devices for moving or lifting great weights by means of
a small force, (2) the construction of engines of war for
throwing projectiles a long distance, (3) the pumping of water
from great depths, (4) the devices of ' wonder-workers '
(Oavfiao-iovpyoi), some depending on pneumatics (like Heron
in the Pneumatica), some using strings, &c, to produce move-
ments like those of living things (like Heron in 'Automata and
Balancings '), some employing floating t bodies (like Archimedes
in ' Floating Bodies '), others using water to measure time
(like Heron in his ' Water-clocks'), and lastly ' sphere-making ',
or the construction of mechanical imitations of the movements
of the heavenly bodies with the uniform circular motion of
water as the motive power. Archimedes, says Pappus, was
held to be the one person who had understood the cause and
the reason of all these various devices, and had applied his
extraordinarily versatile genius and inventiveness to all the
purposes of daily life, and yet, although this brought him
unexampled fame the world over, so that his name was on
every one's lips, he disdained (according to Carpus) to write
any mechanical work save a tract on sphere-making, but
diligently wrote all that he could in a small compass of the
most advanced parts of geometry and of subjects connected
with arithmetic. Carpus himself, says Pappus, as well as
others applied geometry to practical arts, and with reason :
' for geometry is in no wise injured, nay it is by nature
capable of giving substance to many arts by being associated
with them, and, so far from being injured, it may be said,
while itself advancing those arts, to be honoured and adorned
by them in return.'
The object of the Book.
Pappus then describes the object of the Book, namely
to set out the propositions which the ancients established by
geometrical methods, besides certain useful theorems dis-
covered by himself, but in a shorter and clearer form and
430 PAPPUS OF ALEXANDRIA
in better logical sequence than his predecessors had attained.
The sort of questions to be dealt with are (1) a comparison
between the force required to move a given weight along
a horizontal plane and that required to move the same weight
upwards on an inclined plane, (2) the finding of two mean
proportionals between two unequal straight lines, (3) given
a toothed wheel with a certain number of teeth, to find the
diameter of, and to construct, another wheel with a given num-
ber of teeth to work on the former. Each of these things, he says,
will be clearly understood in its proper place if the principles
on which the ' centrobaric doctrine ' is built up are first set out.
It is not necessary, he adds, to define what is meant by ' heavy '
and ' light ' or upward, and downward motion, since these
matters are discussed by Ptolemy in his Mathematical but
the notion of the centre of gravity is so fundamental in the
whole theory of mechanics that it is essential in the first
place to explain what is meant by the ' centre of gravity '
of any body.
On the centre of gravity.
Pappus then defines the centre of gravity as ' the point
within a body which is such that, if the weight be conceived
to be suspended from the point, it will remain at rest in any
position in which it is put '} The method of determining the
point by means of the intersection, first of planes, and then of
straight lines, is next explained (chaps. 1,2), and Pappus then
proves (Prop. 2) a proposition of some difficulty, namely that,
if D, E, F be points on the sides BG, GA, AB of a triangle ABG
such that BD:DC= GE:EA = AF:FB,
then the centre of gravity of the triangle ABG is also the
centre of gravity of the triangle DEF.
Let H, K be the middle points of BG, GA respectively;
join AH, BK. Join EK meeting DE in L.
Then AH, BK meet in G, the centre of gravity of the
triangle ABG, and AG = 2 GH, BG = 2 BK, so that .
GA :AK = AB:HK = BG: GK = AG : GH.
1 Pappus, viii, p. 1030. 11-13.
THE COLLECTION. BOOK VIII
Now, by hypothesis,
GE:EA =BD:DC,
whence CA : AE = BC : CD,
and, if we halve the antecedents,
AK:AE= HC:CD;
therefore AK : EK = HC : HD or BE : HD,
431
whence, componendo, CE : T^A" = 5i) : DH. (1)
But AF: FB= BD: DC = (J5D : 2)//) . (DH : DC)
= (CE:EK).(DH:DC). (2)
Now, i£XZ) being a transversal cutting the sides of the
triangle KHC, we have
HL:KL = (CE:EK) . (DH : DC). (3)
[This is ' Menelaus's theorem ' ; Pappus does not, however,
quote it, but proves the relation ad hoc in an added lemma by
drawing CM parallel to DE to meet HK produced in M. The
proof is easy, for HL . LK = ( HL . LM) (Zif . LK)
= (HD:DC).(CE:EK).]
It follows from (2) and (3) that
AF: FB = HL: LK,
and, since A B is parallel to HK, and AH, BK are straight
lines meeting in G, FGL is a straight line.
[This is proved in another easy lemma by reductio ad
absurdum.]
432 PAPPUS OF ALEXANDRIA
We have next to prove that EL = LB.
Now [again by ' Menelaus's theorem ', proved ad hoc by
drawing GN parallel to HK to meet ED produced in N~\
EL:LD = (EK : KG) . (CH : HD). (4)
But, by (1) above, CE.EK = BD:DH;
therefore GK : KE = BH : HD = CH : HD,
so that (EK:KC).(CH:HD)=1, and therefore, from (4),
EL = LD.
It remains to prove that FG = 2GL, which is obvious by
parallels, since FG : GL = AG : GH =2:1.
Two more propositions follow with reference to the centre
of gravity. The first is, Given a rectangle with AB, BG as
adjacent sides, to draw from C a straight line meeting the side
opposite BC in a point D such that, if the trapezium ADCB is
hung from the point D, it will rest with AD, BG horizontal.
G
E
B l\
A I
N C
In other words, the centre of gravity must be in DL, drawn
perpendicular to BG. Pappus proves by analysis that
GIj 2 = 3BL 2 , so that the problem is reduced to that of
dividing BG into parts BL, LG such that this relation holds.
The latter problem is solved (Prop. 6) by taking a point,
say X, in GB such that GX = 3 XB, describing a semicircle on
BG as diameter and drawing XY at right angles to BG to
meet the semicircle in F, so that XY' i = ^ s BG 2 , and then
dividing GB at X so that
GL :LB = CX: XY(= i : J*/ 3 = x/3 : 1).
The second proposition is this (Prop. 7). Given two straight
lines AB, AG, and B a fixed point on AB, if GD be drawn
THE COLLECTION. BOOK VIII 433
with its extremities on AC, AB and so that AC : BD is a given
ratio, then the centre of gravity of the triangle ADC will lie
on a straight line.
Take E, the middle point of AC, and Fa, point on BE such
that DF = 2 FE. Also let H be a point on B A such that
BH=2HA. Draw FG parallel to AC.
Then AG = J AD, and AH=^AB;
therefore #G = § 5Z).
Also .TO = § ,4# = § ,4C. Therefore,
since the ratio AC:BD is given, the
ratio GH: GF is given.
And the angle FGH (= A) is given ;
therefore the triangle FGH is given in
species, and consequently the angle GHF
is given. And if is a given point. *
Therefore HF is a given straight line, and it contains the
centre of gravity of the triangle A DC.
The inclined plane,.
Prop. 8 is on the construction of a plane at a given inclina-
tion to another plane parallel to the horizon, and with this
Pappus leaves theory and proceeds to the practical part.
Prop. 9 (p. 1054. 4 sq.) investigates the problem 'Given
a weight which can be drawn along a plane parallel to the
horizon by a given force; and a plane inclined to the horizon
at a given angle, to find the force required to draw the weight
upwards on the. inclined plane'. This seems to be the first
or only attempt in ancient times to investigate motion on
an inclined plane, and as such it is curious, though of no
value.
Let A be the weight which can be moved by a force C along
a horizontal plane. Conceive a sphere with weight equal to A
placed in contact at L with the given inclined plane ; the circle
OGL represents a section of the sphere by a vertical plane
passing through E its centre and LK the line of greatest slope
drawn through the point L. Draw EG H horizontal and there-
fore parallel to MN in the plane of section, and draw LF
perpendicular to EH. Pappus seems to regard the plane
as rough, since he proceeds to make a system in equilibrium
1523.2 Y f
434 PAPPUS OF ALEXANDRIA
about FL as if L were the fulcrum of a lever. Now the
weight A acts vertically downwards along a straight line
through E. To balance it, Pappus supposes a weight B
attached with its centre of gravity at G.
Then A:B=GF:EF
= (EL-EF):EF
[= (l-sin0):sin0,
where IKMN = $];
and, since LKMN is given, the ratio EF: EL,
and therefore the ratio (EL-EF) : EF, is
given ; thus B is found.
Now, says Pappus, if D is the force which will move B
along a horizontal plane, as C is the force which will move
A along a horizontal plane, the sum of C and D will be the
force required to move the sphere upwards on the inclined
plane. He takes the particular case where 6 = 60°. Then
sin 6 is approximately y§£ (he evidently uses \ . ff for \ \/3),
and A\B— 16:104.
Suppose, for example, that A = 200 talents; then B is 1300
talents. Suppose further that C is 40 man-power ; then, since
D:C = B: A, D = 260 man-power ; and it will take D + C, or
300 man-power, to move the weight up the plane !
Prop. 10 gives, from Heron's Barulcus, the machine con-
sisting of a pulley, interacting toothed wheels, and a spiral
screw working on the last wheel and turned by a handle ;
Pappus merely alters the proportions of the weight to the
force, and of the diameter of the wheels. At the end of
the chapter (pp. 1070-2) he repeats his construction for the
finding of two mean proportionals.
Construction of a conic through Jive points.
Chaps. 13-17 are more interesting, for they contain the
solution of the problem of constructing a conic through five
given points. The problem arises in this way. Suppose we
are given a broken piece of the surface of a cylindrical column
such that no portion of the circumference of either of its base
THE COLLECTION. BOOK VIII
435
is left intact, and let it be required to find the diameter of
a circular section of the cylinder. We take any two points
A, B on the surface of the fragment and by means of these we
find five points on the surface all lying in one plane section,
in general oblique. This is done by taking n\e different radii
and drawing pairs of circles with A, B as centres and with
each of the five radii successively. These pairs of circles with
equal radii, intersecting at points on the surface, determine
five points on the plane bisecting A B at right angles. The five
points are then represented on any plane by triangulation.
Suppose the points are A, B, C, D, E and are such that
^ no two of the lines connecting the different pairs are parallel.
cr
Q!
This case can be reduced to the construction of a conic through
the five points A, B, D, E, F where EF is parallel to AB.
This is shown in a subsequent lemma (chap. 16).
For, if EF be drawn through E parallel to A B, and if CD
meet AB in and EF in 0', we have, by the well-known
proposition about intersecting chords,
C0.0D:A0.0B = CO' . O'D : EC . O'F,
whence O'F is known, and F is determined.
We have then (Prop. 13) to construct a conic through A, B,
D, E, F, where EF is parallel to AB.
Bisect AB, EF at V, W ; then VW produced both ways
is a diameter. Draw DR, the chord through D parallel
F f 2
436 PAPPUS OF ALEXANDRIA
to this diameter. Then R is determined by means of the
relation
RG.GD.BG.GA = RE.EB:FE.EE (1)
in this way.
Join DB, RA, meeting EF in K, L respectively.
Then, by similar triangles,
RG.GD.BG.GA = (RE : EL) . (BE : EK)
= RE.EB:KE.EL
Therefore, by ( 1 ), FE.EE = KE . EL,
whence EL is determined, and therefore L. The intersection
of AL, BE determines R.
Next, in order to find the extremities P, P / of the diameter
through V, W, we draw ED, RF meeting FP / in M, JS T respec-
tively.
Then, as before,
FW. WE:P'W. WP = FE.EE-.RE.ED, by the ellipse,
= FW.WE-.NW.WM, by similar triangles.
Therefore P' W. WP = NW. WM ;
and similarly we can find the value of P'V . VP.
Now, says Pappus, since P'W. WP and P'V.VP are given
areas and the points V, W are given, P, P' are given. His
determination of P, P' amounts (Prop. 14 following) to an
elimination of one of the points and the finding of the other
by means of an equation of the second degree.
Take two points Q, Q' on the diameter such that
P'V.VP=WV.VQ, (a)
P f W.WP = VW.WQ'\ (13)
Q, Q' are thus known, while P, P' remain to be found.
By (a) P'V: VW= QV: VP,
whence P' W : V W = PQ : P V.
Therefore, by means of (/?),
PQ:PV=Q / W:WP,
THE COLLECTION. BOOK VIII 437
so that PQ:QV=Q'W:PQ' i
or PQ.PQ'=QV.Q'W.
Thus P can be found, and similarly P'.
The conjugate diameter is found by virtue of the relation
(conjugate diam.) 2 : PP' 2 = p : PP'.
where p is the latus rectum to PP f determined by the property
of the curve f , P F = AV'-.PV.VF .
Problem, Given tivo conjugate diameters of an ellipse,
to find the axes.
Lastly, Pappus shows (Prop. 14, chap. 17) how, when we are
given two conjugate diameters, we can find the axes. The
construction is as follows. Let A B, CD be conjugate diameters
(CD being the greater), E the centre.
Produce EA to Hso that
EA.AH=DE 2 .
Through A draw FG parallel to CD. Bisect EH in K, and
draw KL at right angles to EH meeting FG in L.
B
M"\\
V
r* - ^^ P\
^ -
,* ~y
l\
\/>
/ N
J/0
^^.a
/K
F
With L as centre, and LE as radius, describe a circle cutting
GF in G, F.
Join EF, EG, and from A draw AM, AN parallel to EF, EG
respectively.
438
PAPPUS OF ALEXANDRIA
Take points P, R on EG, EF such that
EP 2 = GE. EM, and ER 2 = FE.EN.
Then EP is half the major axis, and ER half the minor axis.
Pappus omits the proof.
Problem of seven hexagons in a circle.
Prop. 19 (chap. 23) is a curious problem. To inscribe seven
equal regular hexagons in a circle in such a way that one
is about the centre of the circle, while six others stand on its
sides and have the opposite sides in each case placed as chords
in the circle.
Suppose GHKLNM to be the hexagon so described on HK,
a side of the inner hexagon ; OKL will then be a straight line.
Produce OL to meet the circle in P.
Then OK = KL = LN. Therefore, in the triangle OLN,
OL - 2LN, while the included angle OLN (— 120°) is also
given. Therefore the triangle is given in species; therefore
the ratio ON : NL is given, and, since ON is given, the side NL
of each of the hexagons is given.
Pappus gives the auxiliary construction thus. Let AF be
taken equal to the radius OP. Let AC — \AF, and on A as
base describe a segment of a circle containing an angle of 60°.
Take GE equal to § AC, and draw EB to touch the circle at B.
THE COLLECTION. BOOK VIII 439
Then he proves that, if we join A B, A B is equal to the length
of the side of the hexagon required.
Produce BC to D so that BD = BA, and join DA. ABD
is then equilateral.
Since EB is a tangent to the segment, AE.EC — EB 2 or
AE: EB = EB : EC, and the triangles EAB, EBC are similar.
Therefore BA 2 : BC 2 = AE 2 : EB* = AE'.EC = 9 : 4 ;
and BC = %BA = §52), so that £6' = 2 CD.
But Ci^= 2C.4 ; therefore AC:CF= DC:CB, and 47), BF
are parallel.
Therefore Itf 7 : AD = BC.CD = 2 : 1, so that
BF=2AD = 2AB.
Also £FBC= A BDA = 60°, so that ZARF= 120°, and
the triangle J.I?i^is therefore equal and similar to the required
triangle NLO.
Construction of toothed ivheels and indented screws.
The rest of the Book is devoted to the construction (1) of
toothed wheels with a given number of teeth equal to those of
a given wheel, (2) of a cylindrical helix, the cochlias, indented
so* as to work on a toothed wheel. The text is evidently
defective, and at the end an interpolator has inserted extracts
about the mechanical powers from Heron's Mechanics.
XX
ALGEBRA: DIOPHANTUS OF ALEXANDRIA
Beginnings learnt from Egypt.
In algebra, as in geometry, the Greeks learnt the beginnings
from the Egyptians. Familiarity on the part of the Greeks
with Egyptian methods of calculation is well attested. (1)
These methods are found in operation in the Heronian writings
and collections. (2) Psellus in the letter published by Tannery
in his edition of Diophantus speaks of ' the method of arith-
metical calculations used by the Egyptians, by which problems
in analysis are handled ' ; he adds details, doubtless taken
from Anatolius, of the technical terms used for different kinds
of numbers, including the powers of the unknown quantity.
(3) The scholiast to Plato's Charmides 165 E says that 'parts
of XoyiarTiKtj, the science of calculation, are the so-called Greek
and Egyptian methods in multiplications and divisions, and
the additions and subtractions of fractions '. (4) Plato himself
in the Laws 819 A-c says that free-born boys should, as is the
practice in Egypt, learn, side by side with reading, simple
mathematical calculations adapted to their age, which should
be put into a form such as to combine amusement with
instruction : problems about the distribution of, say, apples or
garlands, the calculation of mixtures, and other questions
arising in military or civil life.
' Hau '-calculations.
r
The Egyptian calculations here in point (apart from their
method of writing and calculating in fractions, which, with
the exception of §, were always decomposed and written
as the sum of a diminishing series of aliquot parts or sub-
multiples) are the /iau-calculations. Hau, meaning a heap, is
the term denoting the unknown quantity, and the calculations
' HAU '-CALCULATIONS 441
in terms of it are equivalent to the solutions of simple equations
with one unknown quantity. Examples from the Papyrus
Rhind correspond to the following equations :
_2 />» l jL /yt I 1 rp _L ryi Q O
■3 iAy T^ o *&/ i^ rj tAj ^ iAs — O O ,
(a? + §a?)--|(a; + §a?) = 10.
The Egyptians anticipated, though only in an elementary
form, a favourite method of Diophantus, that of the ' false
supposition ' or ' regula falsi \ An arbitrary assumption is
made as to the value of the unknown, and the true value
is afterwards found by a comparison of the result of sub-
stituting the wrong value in the original expression with the
actual data. Two examples may be given. The first, from
the Papyrus Rhind, is the problem of dividing 100 loaves
among five persons in such a way that the shares are in
arithmetical progression, and one-seventh of the sum of the
first three shares is equal to the sum of the other two. If
a + 4;d, a+3d, a + 2d, a + d, a be the shares, then
Sa + 9d = 7(2a + d),
or d = 5ja.
Ahmes says, without any explanation, ' make the difference,
as it is, 5-J', and then, assuming a = 1, writes the series
23, 17},' 12, 6£, 1. The addition of these gives 60, and 100 is
If times 60. Ahmes says simply 'multiply If times' and
thus gets the correct values 38|, 29f, 20, 10§|, 1|.
The second example (taken from the Berlin Papyrus 6619)
is the solution of the equations
x 2 +y 2 = 100, •
x :y = 1 :*|, or y = \x.
x is first assumed to be 1 , and x 2 + y 2 is thus found to be f | .
In order to make 100, f§ has to be multiplied by 64 or 8 2 .
The true value of x is therefore 8 times 1 , or 8.
Arithmetical epigrams in the Greek Anthology.
The simple equations eolved in the Papyrus Rhind are just
the kind of equations of which we find many examples in the
442 ALGEBRA: DIOPHANTUS OF ALEXANDRIA
arithmetical epigrams contained in the Greek Anthology. Most
of these appear under the name of Metrodorus, a grammarian,
probably of the time of the Emperors Anastasius I (a.d. 491-
518) and Justin I (a.d. 518-27). They were obviously only
collected by Metrodorus, from ancient as well as more recent
sources. Many of the epigrams (46 in number) lead to simple
equations, and several of them are problems of dividing a num-
ber of apples or nuts among a certain number of persons, that
is to say, the very type of problem mentioned by Plato. For
example, a number of apples has to be determined such that,
if four persons out of six receive one-third, one-eighth, one-
fourth and one-fifth respectively of the whole number, while
the fifth person receives 1 apples, there is one apple left over
for the sixth person, i.e.
2;X + ±X + %x + ~x + 10 + 1 — x.
Just as Plato alludes to bowls ((f>id\ai) of different metals,
there are problems in which the weights of bowls have to
be found. We are thus enabled to understand the allusions of
Proclus and the scholiast on Charmides 165 E to fi-qXiTai
and (jytaXiraL dpi6/xoi, 'numbers of apples or of bowls'.
It is evident from Plato's allusions that the origin of such
simple algebraical problems dates back, at least, to the fifth
century B.C.
The following is a classification of the problems in the
Anthology. (1) Twenty- three are simple equations in one
unknown and of the type shown above; one of these is an
epigram on the age of Diophantus and certain incidents of
his life (xiv. 126). (2) Twelve are easy simultaneous equations
with two unknowns, like Dioph. I. 6 ; they can of course be
reduced to a simple equation with one unknown by means of
an easy elimination. One other (xiv. 51) gives simultaneous
equations in three unknowns
# = 2/ + §z, y = * + £«% z=10+§2/>
and one (xiv. 49) gives four equations in four unknowns,
x + y = 40, x + z=45, x + u = 36, x + y + z + u = 60.
With these may be compared Dioph. I. 16-21, as well as the
general solution of any number of simultaneous linear equa-
EPIGRAMS IN THE GREEK ANTHOLOGY 443
fcions of this type with the same number of unknown quantities
which was given by Thymaridas, an early Pythagorean, and
was called the e7rdu0rj/xa, ' flower ' or ' bloom ' of Thymaridas
(see vol. i, pp. 94-6). (3) Six more are problems of the usual
type about the filling and emptying of vessels by pipes ; e.g.
(xiv. 130) one pipe fills the vessel in one day, a second in two
and a third in three ; how long will all three running together
take to fill it? Another about brickmakers (xiv. 136) is of
the same sort.
Indeterminate equations of the first degree.
The Anthology contains (4) two indeterminate equations of
the first degree which can be solved in positive integers in an
infinite number of ways (xiv. 48, 144) ; the first is a distribu-
tion of apples, 3x in number, into parts satisfying the equation
x — Sy = y t where y is not less than 2 ; the second leads to
three equations connecting four unknown quantities :
x + y = x x + y u
x= 2y v
x 1 = 3y,
the general solution of which is x = 4&, y — k, x x = 3 k,
y 1 = 2 k. These very equations, which, however, are made
determinate by assuming that x + y = x 1 + y 1 — 100, are solved
in Dioph. I. 12.
Enough has been said to show that Diophantus was not
the inventor of Algebra. Nor was he the first to solve inde-
terminate problems of the second degree.
Indeterminate equations of second degree before
Diophantus.
Take first the problem (Dioph. II. 8) of dividing a square
number into two squares, or of finding a right-angled triangle
with sides in rational numbers. We have already seen that
Pythagoras is credited with the discovery of a general formula
for finding such triangles, namely,
^ +{ 1( 71 2_ 1)} 2 ={ 1^2 +1)}2j
444 ALGEBRA: DIOPHANTUS OF ALEXANDRIA
where n is any odd number, and Plato with another formula
of the same sort, namely (2n) 2 + (n 2 — l) 2 = (n 2 +\) 2 . Euclid
(Lemma following X. 28) finds the following more general
formula
m 2 u 2 p 2 q 2 = { \ (mnp 2 + mnq 2 ) } 2 — { \ (mnp 2 — mnq 2 ) } 2 .
The Pythagoreans too, as we have seen (vol. i, pp. 91-3),
solved another indeterminate problem, discovering, by means
of the series of ' side- ' and ' diameter-numbers ', any number
of successive integral solutions of the equations
2x 2 — y 2 = + 1.
Diophantus does not particularly mention this equation,
but from the Lemma to VI. 15 it is clear that he knew how
to find any number of solutions when one is known. Thus,
seeing that 2 x 2 — 1 = y 2 is satisfied by x = 1 , y == 1 , he would
put
2 ( 1 + x) 2 — 1 = a. square
= (px-1) 2 , say;
whence x = (4 + 2p)/(p 2 -2).
Take the value p — 2, and we have x = 4, and x+ I = 5 ;
in this case 2 . 5 2 — 1 = 49 = 7 2 . Putting x + 5 in place of x,
we can find a still higher value, and so on.
Indeterminate equations in the Heronian collections.
Some further Greek examples of indeterminate analysis are
now available. They come from the Constantinople manuscript
(probably of the twelfth century) from which Schone edited
the Metrica of Heron ; they have been published and translated
by Heiberg, with comments by Zeuthen. 1 Two of the problems
(thirteen in number) had been published in a less complete
form in Hultsch's Heron (Geeponicus, 78, 79) ; the others
are new.
I. The first problem is to find two rectangles such that the
perimeter of the second is three times that of the first, and
the area of the first is three times that of the second. The
1 Biblioiheca mathematlca, viii s , 1907-8, pp. 118-34. See now Geom.
2A. 1-13 in Heron, vol. iv (ed. Heiberg), pp. 414-26.
HERONIAN INDETERMINATE EQUATIONS 445
number 3 is of course only an illustration, and the problem is
equivalent to the solution of the equations
(1) u + v — n(x + y))
(2) xy = n.uv )
The solution given in the text is equivalent to
x=2n 3 — 1, y — 2ti 3,
u — n(4n 3 — 2), v — n
Zeuthen suggests that the solution may have been obtained
thus. As the problem is indeterminate, it would be natural
to start with some hypothesis, e.g. to put v = n. It would
follow from equation (1) that u is a multiple of n, say nz.
We have then
x + y = 1 + z,
while, by (2), xy — n 3 z,
whence xy = n 3 (x + y) — n 3 ,
or (x — n 3 ) (y — n 3 ) = n 3 (n 3 — 1 ).
An obvious solution is
x — n 3 = n 3 — 1, y — n 3 — it 3 ,
which gives z — 2 n 3 — 1 + 2 ^ :i — 1 = 4?i 3 — 2, so that
u = ttig — 7i (4 ii 3 — 2).
II. The second is a similar problem about two rectangles,
equivalent to the solution of the equations
(1) x + y = u + v |
(2) or?/ = 7i . m?J
and the solution given in the text is
x + y = u + v = n 3 — 1, (3)
u=n— 1, v = 7i (^ 2 — 1)^
a> = n 2 -l, 2/==7i 2 (7i~l))
In this case trial may have been made of the assumptions
v = nx, y = n 2 u,
446 ALGEBRA: DIOPHANTUS OF ALEXANDRIA
when equation (1) would give
(n — \)x = (ri z — l)u,
a solution of which is x = n 2 — 1, u = n — 1 .
III. The fifth problem is interesting in one respect. We are
asked to find a right-angled triangle (in rational numbers)
with area of 5 feet. We are told to multiply 5 by some
square containing 6 as a factor, e.g. 36. This makes 180,
and this is the area of the triangle (9, 40, 41). Dividing each
sicle by 6, we have the triangle required. The author, then,
is aware that the area of a right-angled triangle with sides in
whole numbers is divisible by 6. If we take the Euclidean
formula for a right-angled triangle, making the sides a . mn,
a . ^(m 2 — n 2 ), a . %(m 2 + n 2 ), where a is any number, and m, n
are numbers which are both odd or both even, the area is
\mn (m — n) (m + n)a 2 ,
and, as a matter of fact, the number mn(m — ri)(m + ri) is
divisible by 24, as was proved later (for another purpose) by
Leonardo of Pisa.
IV. The last four problems (10 to 13) are of great interest.
They are different particular cases of one problem, that of
finding a rational right-angled triangle such that the numerical
sum of its area and its perimeter is a given number. The
author's solution depends on the following formulae, where
a, b are the perpendiculars, and c the hypotenuse, of a right-
angled triangle, S its area, r the radius of the inscribed circle,
and s = %(a + b + c);
S = rs = %ab, r + s = a + b, c — s — r.
(The proof of these formulae by means of the usual figure,
namely that used by Heron to prove the formula
S = V{s(s-a)(s-b)(s-c)},
is easy.)
Solving the first two equations, in order to find a and b,
we have
a \ = ±[r + s+ V{(r + s) 2 -8rs]],
which formula is actually used by the author for finding a
HERONIAN INDETERMINATE EQUATIONS 447
and b. The method employed is to take the sum of the area
and the perimeter S + 2 s, separated into its two obvious
factors s(r+2), to put s(r+2) = A (the given number), and
then to separate A into suitable factors to which s and r + 2
may be equated. They must obviously be such that sr, the
area, is divisible by 6. To take the first example where
A = 280 :' the possible factors are 2 x 140, 4 x 70, 5 x 56, 7 x 40,
8 x 35, 10 x 28, 14 x 20. The suitable factors in this case are
r+2 = 8, s = 35, because r is then equal to 6, and rs is
a multiple of 6.
The author then says that
a= \ [6 + 35- \/ {(6 + 35) 2 -8. 6. 35}] = £(41-1) = 20,
6 = £(41 + 1)=21,
c = 35-6 = 29.
The triangle is« therefore (20, 21, 29) in this case. The
triangles found in the other three cases, by the same method,
are (9, 40, 41), (8, 15, 17) and (9, 12, 15).
Unfortunately there is no guide to the date of the problems
just given. The probability is that the original formulation
of the most important of the problems belongs to the period
between Euclid and Diophantus. This supposition best agrees
with the fact that the problems include nothing taken from
the great collection in the Arithmetica. On the other hand,
it is strange that none of the seven problems above mentioned
is found in Diophantus. The five relating to rational right-
angled triangles might well have been included by him ; thus
he finds rational right-angled triangles such that the area plus
or minus one of the perpendiculars is a given number, but not
the rational triangle which has a given area ; and he finds
rational right-angled triangles such that the area plus or minus
the sum of two sides is a given number, but not the rational
triangle such that the sum of the area and the three sides is
a given number. The omitted problems might, it is true, have
come in the lost Books ; but, on the other hand, Book VI would
have been the appropriate place for them.
The crowning example of a difficult indeterminate problem
propounded before Diophantus's time is the Cattle-Problem
attributed to Archimedes, described above (pp. 97-8).
448 ALGEBRA: DIOPHANTUS OF ALEXANDRIA
Numerical solution of quadratic equations.
The geometrical algebra of the Greeks has been in evidence
all through our history from the -Pythagoreans downwards,
and no more need be said of it here except that its arithmetical
application was no new thing in Diophantus. It is probable,
for example, that the solution of the quadratic equation,
discovered first by geometry, was applied for the purpose of
finding numerical values for the unknown as early as Euclid,
if not earlier still. In Heron the numerical solution of
equations is well established, so that Diophantus was not the
first to treat equations algebraically. What he did was to
take a step forward towards an algebraic notation.
The date of Diophantus can now be fixed with fair certainty.
He was later than Hypsicles, from whom he quotes a definition
of a polygonal number, and earlier than Theon of Alexandria,
who has a quotation from Diophantus's definitions. The
possible limits of date are therefore, say, 150 B.C. to A.D. 350.
But the letter of Psellus already mentioned says that Anatolius
(Bishop of Laodicea about a.d. 280) dedicated to Diophantus
a concise treatise on the Egyptian method of reckoning ;
hence Diophantus must have been a contemporary, so that he
probably flourished A.D. 250 or not much later.
An epigram in the Anthology gives some personal particulars :
his boyhood lasted Jth of his life ; his beard grew after x^th
more ; he married after -|th more, and his son was born 5 years
later ; the son lived to half his father's age, and the father
died 4 years after his son. Thus, if x was his age when
he died,
which gives x = 84.
Works of Diophantus.
The works on which the fame of Diophantus rests are :
(1) the Arithmetica (originally in thirteen Books),
(2) a tract On Polygonal Numbers.
WORKS 44 ( J
Six Books only of the former and a fragment of the latter
survive.
Allusions in the Arithmetica imply the existence of
(3) A collection of propositions under the title of Porisms ;
in three propositions (3, 5, 16) of Book V, Diophantus quotes
as known certain propositions in the Theory of Numbers,
prefixing to the statement of them the words ' We have it in
the Porisms that . . .'
A scholium on a passage of Iamblichus, where Iamblichus
cites a dictum of certain Pythagoreans about the unit being
the dividing line (fieOopLov) between number and aliquot parts,
says ' thus Diophantus in the Moriastica .... for he describes
as " parts " the progression without limit in the direction of
less than the unit '. The Moriastica may be a separate work
by Diophantus giving rules for reckoning with fractions ; but
I do not feel sure that the reference may not simply be to the
definitions at the beginning of the Arithmetica.
The Arithmetica.
The seven lost Books and their place.
None of the manuscripts which we possess contain more
than six Books of the Arithmetica, the only variations being
that some few divide the six Books into seven, while one or
two give the fragment on Polygonal Numbers as VIII. The
missing Books were evidently lost at a very early date.
Tannery suggests that Hypatia's commentary extended only
to the first six Books, and that she left untouched the remain-
ing seven, which, partly as a consequence, were first forgotten
and then lost (cf. the case of Apollonius's Conies, where the
only Books which have survived in Greek, I-IV, are those
on which Eutocius commented). There is no sign that even
the Arabians ever possessed the missing Books. The Fakhra,
an algebraical treatise by Abu Bekr Muh. b. al-Hasan al-
Karkhi (d. about 1029), contains a collection of problems in
determinate and indeterminate analysis which not only show
that their author had deeply studied Diophantus but in many
cases are taken direct from the Arithmetica, sometimes with
a change in constants; in the fourth section of the work,
1523.2 Q g
450 DIOPHANTUS OF ALEXANDRIA
between problems corresponding to problems in Dioph. II
and III, are 25 problems not found in Diophantus, but
internal evidence, and especially the admission of irrational
results (which are always avoided by Diophantus), exclude
the hypothesis that we have here one of the lost Books.
Nor is there any sign that more of the work than we possess
was known to Abul Wafa al-Buzjani (a.d. 940-98) who wrote
a ' commentary on the algebra of Diophantus ', as well as
a ' Book of proofs of propositions used by Diophantus in his
work'. These facts again point to the conclusion that the
lost Books were lost before the tenth century.
The old view of the place originally occupied by the lost
seven Books is that of Nesselmann, who argued it with great
ability. 1 According to him (1) much less of Diophantus is
wanting than would naturally be supposed on the basis of
the numerical proportion of 7 lost to 6 extant Books, (2) the
missing portion came, not at the end, but in the middle of
the work, and indeed mostly between the first and second
Books. Nesselmann's general argument is that, if we care-
fully read the last four Books, from the third to the sixth,
we shall find that Diophantus moves in a rigidly defined and
limited circle of methods and artifices, and seems in fact to be
at the end of his resources. As regards the possible contents
of the lost portion on this hypothesis, Nesselmann can only
point to (1) topics which we should expect to find treated,
either because foreshadowed by the author himself or as
necessary for the elucidation or completion of the whole
subject, (2) the Porisms; under head (l) come, (a) deter-
minate equations of the second degree, and (6) indeterminate
equations of the first degree. Diophantus does indeed promise
to show how to solve the general quadratic ax 2 ± bx ± c = so
far as it has rational and positive solutions ; the suitable place
for this would have been between Books I and II. But there
is nothing whatever to show that indeterminate equations
of the first degree formed part of the writer's plan. Hence
Nesselmann is far from accounting for the contents of seven
whole Books ; and he is forced to the conjecture that the six
Books may originally have been divided into even more than
seven Books ; there is, however, no evidence to support this.
1 Nesselmann, Algebra der Griechen, pp. 264-73.
RELATION OF WORKS 451
Relation of the 'Porisms' to the Arithmetica.
Did the Porisms form part of the Arithmetica in its original
form ? The phrase in which they are alluded to, and which
occurs three times, ' We have it in the Poi^isms that . . .' suggests
that they were a distinct collection of propositions concerning
the properties of certain numbers, their divisibility into a
certain number of squares, and so on ; and it is possible that
it was from the same collection that Diophantus took the
numerous other propositions which he assumes, explicitly or
implicitly. If the collection was part of the Arithmetica, it
would be strange to quote the propositions under a separate
title ' The Porisms ' when it would have been more natural
to refer to particular propositions of particular Books, and
more natural still to say tovto yap irpoSiSeiKrai, or some such
phrase, ' for this has been proved ', without any reference to
the particular place where the proof occurred. The expression
'We have it in the Porisms ' (in the plural) would be still
more inappropriate if the Porisms had been, as Tannery
supposed, not collected together as one or more Books of the
Arithmetica, but scattered about in the work as corollaries to
particular propositions. Hence I agree with the view of
Hultsch that the Porisms were not included in the Arith-
metica at all, but formed a separate work.
If this is right, we cannot any longer hold to the view of
Nesselmann that the lost Books were in the middle and not at
the end of the treatise ; indeed Tannery produces strong
arguments in favour of the contrary view, that it is the last
and most difficult Books which are lost. He replies first to
the assumption that Diophantus could not have proceeded 3
to problems more difficult than those of Book V. 'If the
fifth or the sixth Book of the Arithmetica had been lost, who,
pray, among us would have believed that such problems had
ever been attempted by the Greeks 1 It would be the greatest
error, in any case in which a thing cannot clearly be proved
to have been unknown to all the ancients, to maintain that
it could not have been known to some Greek mathematician.
If we do not know to what lengths Archimedes brought the
theory of numbers (to say nothing of other things), let us
admit our ignorance. But, between the famous problem of the
Gg2
452 DIOPHANTUS OF ALEXANDRIA
cattle and the most difficult of Diophantus's problems, is there
not a sufficient gap to require seven Books to fill if? And,
without attributing to the ancients what modern mathe-
maticians have discovered, may not a number of the things
attributed to the Indians and Arabs have been drawn from
Greek sources? May not the same be said of a problem
solved by Leonardo of Pisa, which is very similar to those of
Diophantus but is not now to be found in the Arithmetica 1
In fact, it may fairly be said that, when Chasles made his
reasonably probable restitution of the Porisms of Euclid, he,
notwithstanding that he had Pappus's lemmas to help him,
undertook a more difficult task than he would have undertaken
if he had attempted to fill up seven Diophantine Books with
numerical problems which the Greeks may reasonably be
supposed to have solved.' *
It is not so easy to agree with Tannery's view of the relation
of the treatise On Polygonal Numbers to the Arithmetica.
According to him, just as Serenus's treatise on the sections
of cones and cylinders was added to the mutilated Conies of
Apollonius consisting of four Books only, in order to make up
a convenient volume, so the tract on Polygonal Numbers was
added to the remains of the Arithmetica, though forming no
part of the larger work. 2 Thus Tannery would seem to deny
the genuineness of the whole tract on Polygonal Numbers,
though in his text he only signalizes the portion beginning
with the enunciation of the problem ' Given a number, to find
in how many ways it can be a polygonal number ' as ' a vain
attempt by a commentator ' to solve this problem. Hultsch,
on the other hand, thinks that we may conclude that Dio-
phantus really solved the problem. The tract begins, like
Book I of the Arithmetica, with definitions and preliminary
propositions ; then comes the difficult problem quoted, the
discussion of which breaks off in our text after a few pages,
and to these it would be easy to tack on a great variety of
other problems.
The name of Diophantus was used, as were the names of
Euclid, Archimedes and Heron in their turn, for the pur-
pose of palming off the compilations of much later authors.
1 Diophantus, ed. Tannery, vol. ii, p. xx.
2 lb., p. xviii.
RELATION OF WORKS 453
Tannery includes in his edition three fragments under the
heading ' Diophantus Pseudepigraphus '. The first, which is
not ' from the Arithmetic of Diophantus ' as its heading states,
is worth notice as containing some particulars of one of ' two
methods of finding the square root of any square number ' ;
we are told to begin by writing the number ' according to
the arrangement of the Indian method', i.e. in the Indian
numerical notation which reached us through the Arabs. The
second fragment is the work edited by C. Henry in 1879 as
Opuscvdum de multiplication et divisione sexagesimalibus
Diophanto vet Pappo attribuendum. The third, beginning
with Aio(j)civTov €7rnre8ofX€TpiKd is a Byzantine compilation
from later reproductions of the y€(ojj.€Tpov/ji€ua and o-repeo-
fMerpovfieva of Heron. Not one of the three fragments has
anything to do with Diophantus.
Commentators from Hypatia dowmvards.
The first commentator on Diophantus of whom we hear
is Hypatia, the daughter of Theon of Alexandria ; she
was murdered by Christian fanatics in a.d. 415. I have
already mentioned the attractive hypothesis of Tannery that
Hypatia's commentary extended only to our six Books, and
that this accounts for their survival when the rest were lost.
It is possible that the remarks of Psellus (eleventh century) at
the beginning of his letter about Diophantus, Anatolius and
the Egyptian method of arithmetical reckoning were taken
from Hypatia's commentary.
Georgius Pachymeres (1240 to about 1310) wrote in Greek
a paraphrase of at least a portion of Diophantus. Sections
25-44 of this commentary relating to Book I, Def. 1 to Prop.
11, survive. Maximus Planudes (about 1260-1310) also wrote
a systematic commentary on Books I, II. Arabian commen-
tators were Abu'l Wafa al-Buzjani (940-98), Qusta b. Luqa
al-Ba f labakki (d. about 912) and probably Ibn al-Haitham
(about 965-1039).
Translations and editions.
To Regiomontanus belongs the credit of being the first to
call attention to the work of Diophantus as being extant in
454 DIOPHANTUS OF ALEXANDRIA
Greek. In an ratio delivered at the end of: 1463 as an
introduction to a course of lectures on astronomy. which he
gave at Padua in 1403-4 he observed: 'No one has yet
translated from the Greek into Latin the fine thirteen Books
of Diophantus, in which the very flower of the whole of
arithmetic lies hid, the ars rei et census which to-day they
call by the Arabic name of Algebra.' Again, in a letter dated
February 5, 1464, to Bianchini, he writes that he has found at
Venice ' Diofantus, a Greek arithmetician who has not yet
been translated into Latin '. Rafael Bombelli was the first to
find a manuscript in the Vatican and to conceive the idea of
publishing the work; this was towards 1570, and, with
Antonio Maria Pazzi, he translated five Books out of the
seven into which the manuscript was divided. The translation
was not published, but Bombelli took all the problems of the
first four Books and some of those of the fifth and embodied
them in his Algebra (1572), interspersing them with his own
problems.
The next writer on Diophantus was Wilhelm Holzmann,
who called himself Xylander, and who with extraordinary
industry and care produced a very meritorious Latin trans-
lation with commentary (1575). Xylander was an enthusiast
for Diophantus, and his preface and notes are often delightful
reading. Unfortunately the book is now very rare. The
standard edition of Diophantus till recent years was that of
Bachet, who in 1621 published for the first time the Greek
text with Latin translation and notes. A second edition
(1670) was carelessly printed and is untrustworthy as regards
the text ; on the other hand it contained the epoch-making
notes of Fermat ; the editor was S. Fermat, his son. The
great blot on the work of Bachet is his attitude to Xylander,
to whose translation he owed more than he was willing to
avow. Unfortunately neither Bachet nor Xylander was able
to use the best manuscripts ; that used by Bachet was Parisinus
2379 (of the middle of the sixteenth century), with the help
of a transcription of part of a Vatican MS. (Vat. gr. 304 of
the sixteenth century), while Xylander's manuscript was the
Wolfenbuttel MS. Guelferbytanus Gudianus 1 (fifteenth cen-
tury). The best and most ancient manuscript is that of
Madrid (Matritensis 48 of the thirteenth century) which was
TRANSLATIONS AND EDITIONS 455
unfortunately spoiled by corrections made, especially in Books
I, II, from some manuscript of the ' Planudean ' class ; where
this is the case recourse must be had to Vat. gr. 191 which
was copied from it before it had suffered the general alteration
referred to : these are the first two of the manuscripts used by
Tannery in his definitive ' edition of the Greek text (Teubner,
1893, 1895).
Other editors can only be shortly enumerated. In 1585
Simon Stevin published a French version of the first four
Books, based on Xylander. Albert Girard added the fifth and
sixth Books, the complete edition appearing in 1625. German
translations were brought out by Otto Schulz in 1822 and by
G. Wertheim in 1890. Poselger translated the fragment on
Polygonal Numbers in 1810. All these translations depended
on the text of Bachet.
A reproduction of Diophantus in modern notation with
introduction and notes by the present writer (second edition
1910) is based on the text of Tannery and may claim to be the
most complete and up-to-date edition.
My account of the Arithmetica of Diophantus will be most
conveniently arranged under three main headings (1) the
notation and definitions, (2) the principal methods employed,
so far as they can be generally stated, (3) the nature of the
contents, including the assumed Porisms, with indications of
the devices by which the problems are solved.
Notation and definitions.
In his work Die Algebra der Griechen Nesselmann distin-
guishes three stages in the evolution of algebra. (1) The
first stage he calls ' Rhetorical" Algebra ' or reckoning by
means of complete words. The characteristic of this stage
is the absolute want of all symbols, the whole of the calcula-
tion being carried on by means of complete words and forming
in fact continuous prose. This first stage is represented by
such writers as Iamblichus, all Arabian and Persian algebraists,
and the oldest Italian algebraists and their followers, including
Regiomontanus. (2) The second stage Nesselmann calls the
' Syncopated Algebra ', essentially like the first as regards
456 DIOPHANTUS OF ALEXANDRIA
literary style, but marked by the use of certain abbreviational
symbols for constantly recurring quantities and operations.
To this stage belong Diophantus and, after him, all the later
Europeans until about the middle of the seventeenth century
(with the exception of Vieta, who was the first to establish,
under the name of Logistica speciosa, as distinct from Logistica
numerosa, a regular system of reckoning with letters denoting
magnitudes as well as numbers). (3) To the third stage
Nesselmann gives the name of ' Symbolic Algebra ', which
uses a complete system of notation by signs having no visible
connexion with the words or things which they represent,
a complete language of symbols, which entirely supplants the
' rhetorical ' system, it being possible to work out a solution
without using a single word of ordinary language with the
exception of a connecting word or two here and there used for
clearness' sake.
Sign for the unknoivn (= x), and its origin.
Diophantus's system of notation then is merely abbrevia-
tional. We will consider first the representation of the
unknown quantity (our x). Diophantus defines the unknown
quantity as ' containing an indeterminate or undefined multi-
tude of units' .(irXfjOos uoudScou dopicrrov), adding that it is
called dptduos, i.e. number simply, and is denoted by a certain
sign. This sign is then used all through the book. In the
earliest (the Madrid) MS. the sign takes the form ^, in
Marcianus 308 it appears as S. In the printed editions of
Diophantus before Tannery's it was represented by the final
sigma with an accent, y', which is sufficiently like the second
of the two forms. Where the symbol takes the place of
inflected forms dpi0p.6v, dpiQuov, &c, the termination was put
above and to the right of the sign like an exponent, e.g. y x for
dpiOpov as t s for toy, y°° for dpiOuod; the symbol was, in
addition, doubled in the plural cases, thus s? ot , ss™', &c. The
coefficient is expressed by putting the required Greek numeral
immediately after it; thus yy oi la — 11 dpiduoi, equivalent
to 1 1 x, ?' oc = x, and so on. Tannery gives reasons for think-
ing that in the archetype the case-endings did not appear, and
NOTATION AND DEFINITIONS 457
that the sign was not duplicated for the plural, although such
duplication was the practice of the Byzantines. That the
sign was merely an abbreviation for the word dpid/169 and no
algebraical symbol is shown by the fact that it occurs in the
manuscripts for dpiOfios in the ordinary sense as well as for
dpiOfios in the technical sense of the unknown quantity. Nor
is it confined to Diophantus. It appears in more or less
similar forms in the manuscripts of other Greek mathe-
maticians, e.g. in the Bodleian MS. of Euclid (D'Orville 301)
of the ninth century (in the forms 9 99, or as a curved line
similar to the abbreviation for kcli), in the manuscripts of
the Sand -reckoner of Archimedes (in a form approximat-
ing to y), where again there is confusion caused by the
similarity of the signs for dpiOfios and /cat, in a manuscript
of the Geodaesia included in the Heronian collections edited
by Hultsch (where it appears in various forms resembling
sometimes £ sometimes p, sometimes o, and once £, with
case- endings superposed) and in a manuscript of Theon of
Smyrna.
What is the origin of the sign? It is certainly not the
final sigma, as is proved by several of the forms which it
takes. I found that in the Bodleian manuscript of Diophantus
it is written in the form '^4, larger than and quite unlike the
final sigma. This form, combined with the fact that in one
place Xylander's manuscript read ap for the full word, suggested
to me that the sign might be a simple contraction of the first
two letters of dpiBfios. This seemed to be confirmed by
Gardthausen's mention of a contraction for ap, in the form up
occurring in a papyrus of a.d. 154, since the transition to the
form found in the manuscripts of Diophantus might easily
have been made through an intermediate form < p. The loss of
the downward stroke, or of the loop, would give a close
approximation to the forms which we know. This hypothesis
as to the origin of the sign has not, so far as I know, been
improved upon. It has the immense advantage that it makes
the sign for dp 16/169 similar to the signs for the powers of
the unknown, e.g. A Y for Swapis, K Y for icvfios, and to the
o
sign M for the unit, the sole difference being that the two
letters coalesce into one instead of being separate.
458 DIOPHANTUS OF ALEXANDRIA
Signs for the powers of the unknown and their reciprocals.
The powers of the unknown, corresponding to our x 2 , x ?J . . . x 6 ,
are defined and denoted as follows :
x 2 is Svvctfjus and is denoted by A Y ,
x ?> „ kv/3os „ „ „ K Y ,
X 4 ,, BvvajxoSvvajjLLS „ „ A A,
x 5 „ SvvctfioKvPos „ „ AK ,
x G „ KvfioKvffos „ „ „ K K.
Beyond the sixth power Diophantus does not go. It should
be noted that, while the terms from Kvfios onwards may be
used for the powers of any ordinary known number as well as
for the powers of the unknown, Svuafii? is restricted to the
square of the unknown ; wherever a particular square number
is spoken of, the term is reTpdyoovos dptOfio?. The term
SwanoSyvajiis occurs once in another author, namely in the
Metrica of Heron, 1 where it is used for the fourth power of
the side of a triangle.
Diophantus has also terms and signs for the reciprocals of
the various powers of the unknown, i.e. for 1/x, l/x 2 ....
As an aliquot part was ordinarily denoted by the corresponding
numeral sign with an accent, e.g. ■/= J> ia! = tt> Diophantus
has a mark appended to the symbols for x, x 2 . . . to denote the
reciprocals; this, which is used for aliquot parts as well, is
printed by Tannery thus, *. With Diophantus then
dpiOjxoa-Tou, denoted by ?*, is equivalent to l/x,
SwafiocrTOv, „ A „ „ 1 / x 2 ,
and so on.
The coefficient of the term in x, x 2 ... or l/x, l/x 2 ... is
expressed by the ordinary numeral immediately following,
e.g. AK Y /c<r = 26a; 5 , A Y * <ri> = 250 / x 2 .
Diophantus does not need any signs for the operations of
multiplication and division. Addition is indicated by mere
juxtaposition ; thus K Y a A Y iy ? e corresponds to x z + 1 3 a 2 + 5x.
1 Heron, Metrica, p. 48. 11, 19, Sclione.
NOTATION AND DEFINITIONS 459
When there are units in addition, the units are indicated by
o o
the abbreviation M ; thus K Y a A Y iy s e M /3 corresponds to
a 3 +13# 2 + 5.x+2.
*
The sign (A) for minus and its meaning.
For subtraction alone is a sign used. The full term for
wanting is Aen//-*?, as opposed to virap^is, a forthcoming,
which denotes a positive term. The symbol used to indicate
a wanting, corresponding to our sign for minus, is A, which
is described in the text as a ' \jr turned downwards and
truncated ' (¥ eXXnres kcctco vevov). The description is evidently
interpolated, and it is now certain that the sign has nothing
to do with \jr. Nor is it confined to Diophantus, for it appears
in practically the same form in Heron's Metrical where in one
place the reading of the manuscript is uovdScov oS T i'8',
74— y 1 ^. In the manuscripts of Diophantus, when the sign
is resolved by writing the full word instead of it, it is
generally resolved into Xetyfrei, the dative of Xeiyjris, but in
other places the symbol is used instead of parts of the verb
XeiTTeiv, namely Xinwu or Xeiyjras and once even Xtirctxn ;
sometimes X^iyjr^i in the manuscripts is followed by the
accusative, which shows that in these cases the sign was
wrongly resolved. It is therefore a question whether Dio-
phantus himself ever used the dative Xetyei for minus at all.
The use is certainly foreign to classical Greek. Ptolemy has
in two places Xnyjrav and Xdnova-av respectively followed,
properly, by the accusative, and in one case he has to dirb
7-779 TA Xeicpdev vnb rod dirb tt]s ZT (where the meaning is
ZT 2 — TA 2 ). Hence Heron would probably have written a
participle where the T occurs in the expression quoted above,
say ixovdBcov 08 Xeiyjrao-coi' T€<ro-apaKaiSeKaToi>. On the whole,
therefore, it is probable that in Diophantus, and wherever else
it occurred, A is a compendium for the root of the v§rb Xei7T€ii>,
in fact a A with I placed in the middle (cf. A, an abbreviation
for rdXavTov). This is the hypothesis which I put forward
in 1885, and it seems to be confirmed by the fresh evidence
now available as shown above.
1 Heron, Metrica, p. 156. 8, 10.
460 DIOPHANTUS OF ALEXANDRIA
Attached to the definition of minus is the statement that
'a wanting (i.e. a minus) multiplied by a ivanting makes
a forthcoming (i. e. a plus) ; and a wanting (a minus) multi-
plied by a forthcoming (a plus) makes a ivanting (a 'minus) '.
Since Diophantus uses no sign for plus, he has to put all
the positive terms in an expression together and write all the
negative terms together after the sign for minus ; e.g. for
x z — 5# 2 + 8# — l he necessarily writes K a s ?j A A Y e M a.
The Diophantine notation for fractions as well as for large
numbers has been fully explained with many illustrations
in Chapter II above. It is only necessary to add here that,
when the numerator and denominator consist of composite
expressions in terms of the unknown and its powers, he puts
the numerator first followed by e^ ftopico or uopiov and the
denominator.
Thus A Y i M fi$K kv fiopicp A Y A aM^A Y ^
= (60# 2 + 2520)/(a 4 + 900-60a 2 ), [VI. 12]
O o
and A ie A M A$- kv uopicp A Y A a M A9 A A Y i/3
= (15x a -36)/(x* + 36-12x 2 ) [VI. 14].
For a term in an algebraical expression, i.e. a power of x
with a certain coefficient, and the term containing a certain
number of units, Diophantus uses the word eWo?, 'species',
which primarily means the particular power of the variable
without the coefficient. At the end of the definitions he gives
directions for simplifying equations until each side contains
positive terms only, by the addition or subtraction of coeffi-
cients, and by getting rid of the negative terms (which is done
by adding the necessary quantities to both sides) ; the object,
he says, is to reduce the equation until one term only is left
on each side ; ' but ', he adds, ' I will show you later how, in
the case also where two terms are left equal to one term,
such a problem is solved '. We find in fact that, when he has
to solve a quadratic equation, he endeavours by means of
suitable assumptions to reduce it either to a simple equation
or a pure quadratic. The solution of the mixed quadratic
NOTATION AND DEFINITIONS 461
in three terms is clearly assumed in several places of the
Arithmetics, but Diophantus never gives the necessary ex-
planation of this case as promised in the preface.
Before leaving the notation of Diophantus, we may observe
that the form of it limits him to the use of one unknown at
a time. The disadvantage is obvious. For example, where
we can begin with any number of unknown quantities and
gradually eliminate all but one, Diophantus has practically to
perform his eliminations beforehand so as to express every
quantity occurring in the problem in terms of only one
unknown. When he handles problems which are by nature
indeterminate and would lead in our notation to an inde-
terminate equation containing two or three unknowns, he has
to assume for one or other of these some particular number
arbitrarily chosen, the effect being to make the problem
determinate. However, in doing so, Diophantus is careful
to say that we may for such and such a quantity put any
number whatever, say such and such a number; there is
therefore (as a rule) no real loss of generality. The particular
devices by which he contrives to express all his unknowns
in terms of one unknown are extraordinarily various and
clever. He can, of course, use the same variable y in the
same problem with different significations successively, as
when it is necessary in the course of the problem to solve
a subsidiary problem in order to enable him to make the
coefficients of the different terms of expressions in x such
as will answer his purpose and enable the original problem
to be solved. There are, however, two cases, II. 28, 29, where
for the proper working-out of the problem two unknowns are
imperatively necessary. We should of course use x and y;
Diophantus calls the first y as usual; the second, for want
of a term, he agrees to call in the first instance 'one unit',
i.e. 1. Then later, having completed the part of the solution
necessary to find x, he substitutes its value and uses y over
again for what he had originally called 1. That is, he has to
put his finger on the place to which the 1 has passed, so as
to substitute y for it. This is a tour de force in the particular
cases, and would be difficult or impossible in more complicated
problems.
462 DIOPHANTUS ' OF ALEXANDRIA
The methods of Diophantus.
It should be premised that Diophantus will have in his
solutions no numbers whatever except ' rational ' numbers ;
he admits fractional solutions as well as integral, but he
excludes not only surds and imaginary quantities but also
negative quantities. Of a negative quantity per se, i.e. with-
out some greater positive quantity to subtract it from, he
had apparently no conception. Such equations then as lead
to imaginary or negative roots he regards as useless for his
purpose ; the solution is in these cases dSvparos, impossible.
So we find him (V. 2) describing the equation 4 = 4a; + 20 as
droiros, absurd, because it would give x = — 4. He does, it is
true, make occasional use of a quadratic which would give
a root which is positive but a surd, but only for the purpose
of obtaining limits to the root which are integers or numerical
fractions ; he never uses or tries to express the actual root of
such an equation. When therefore he arrives in the course
of solution at an equation which would give an ' irrational '
result, he retraces his steps, finds out how his equation has
arisen, and how he may, by altering the previous work,
substitute for it another which shall give a rational result.
This gives rise in general to a subsidiary problem the solution
of which ensures a rational result for the problem itself.
It is difficult to give a complete account of Diophantus's
methods without setting out the whole book, so great is the
variety of devices and artifices employed in the different
problems. There are, however, a few general methods which
do admit of differentiation and description, and these we pro-
ceed to set out under subjects.
I. Diophantus's treatment of equations.
(A) Determinate equations.
Diophantus solved without difficulty determinate equations
of the first and second degrees ; of a cubic we find only one
example in the Arithmetica, and that is a very special case.
(1) Pure determinate equations.
Diophantus gives a general rule for this case without regard
to degree. We have to take like from like on both sides of an
DETERMINATE EQUATIONS 463
equation and neutralize negative terms by adding to both
sides, then take like from like again, until we have one term
left equal to one term. After these operations have been
performed, the equation (after dividing out, if both sides
contain a power of x, by the lesser power) reduces to Ax m = B,
and is considered solved. Diophantus regards this as giving
one root only, excluding any negative value as ' impossible '.
No equation of the kind is admitted which does not give
a ' rational ' value, integral or fractional. The value x = is
ignored in the case where the degree of the equation is reduced
by dividing out by any power of x.
(2) Mixed quadratic equations.
Diophantus never gives the explanation of the method of
solution which he promises in the preface. That he had
a definite method like that used in the Geometry of Heron
is proved by clear verbal explanations in different propositions.
As he requires the equation to be in the form of two positive
terms being equal to one positive term, the possible forms for
Diophantus are
(a) mx 2 +px = q, (b) mx 2 = px + q, (c) mx 2 + q=px.
It does not appear that Diophantus divided by m in order to
make the first term a square ; rather he multiplied by m for
this purpose. It is clear that he stated the roots in the above
cases in a form equivalent to
/a -hV+ ^(ip 2 + mq) ,,. ip+ </(ip 2 + mq)
m- m
Jj9+ V(ijJ 2 — mq)
(o)
VI
The explanations which show this are to be found in yi. 6,
in IV. 39 and 31, and in V. 10 and VI. 22 respectively. For
example in V.-10 he has the equation \7x 2 + 17 < 72x, and he
says ' Multiply half the coefficient of x into itself and we have
1296; subtract the product of the coefficient of x 2 and the
term in units, or 28§. The remainder is 1007, the square root
of which is not greater than 31. Add half the coefficient of x
and the result is not greater than 67. Divide by the coefficient
of x 2 , and x is not greater than f^.' In IV. 39 he has the
464 DIOPHANTUS OF ALEXANDRIA
equation 2& 2 >6#+18 and says, 'To solve this, take the square
of half the coefficient of x, i.e. 9, and the product of the unit-
term and the coefficient of x 2 , i.e. 36. Adding, we have 45,
the square root of which is not less than 7. Add half the
coefficient of x [and divide by the coefficient of & 2 ] ; whence x
is not less than 5.' In these cases it will be observed that 3 1
and 7 are not accurate limits, but are the nearest integral
limits which will serve his purpose.
Diophantus always uses the positive sign with the radical,
and there has been much discussion as to whether he knew
that a quadratic equation has two roots. The evidence of the
text is inconclusive because his only object, in every case, is to
get one solution ; in some cases the other root would be
negative, and would therefore naturally be ignored as 'absurd'
or ' impossible '. In yet other cases where the second root is
possible it can be shown to be useless from Diophantus's point
of view. For my part, I find it difficult or impossible to
believe that Diophantus was unaware of the existence of two
real roots in such cases. It is so obvious from the geometrical
form of solution based on Eucl. II. 5, 6 and that contained in
Eucl. VI. 27-9; the construction of VI. 28, too, corresponds
in fact to the negative sign before the radical in the case of the
particular equation there solved, while a quite obvious and
slight variation of the construction would give the solution
corresponding to the 'positive sign.
The following particular cases of quadratics occurring in
the Arithmetica may be quoted, with the results stated by
Diophantus.
x 2 ■ 4x — 4 ; therefore x = 2. (IV. 22)
325a; 2 = 3&+18
8ix 2 + 7x = 7
84x 2 —7x — 7
630o; 2 -73x = 6
630ar+73a? = 6
« = * 7 Aor A. (IV. 31)
x = i. (VI. 6)
= 1. (VI. 7)
»=.&• (VI. 9)
x is rational. (VI. 8)
5x < x 2 -60 < 8x; x not < 11 and not > 12. (V. 30)
\7x 2 +\7 < 72&<19a; 2 +19; x not >f| and not <§§. (V. 10)
22£ < x 2 + 60 < 24&; x not < 19 but < 21. (V. 30)
DETERMINATE EQUATIONS 465
In the first and third of the last three cases the limits are not
accurate, but are integral limits which are a fortiori safe.
In the second f § should have been § J, and it would have been
more correct to say that, if x is not greater than f -£ and not
less than ff , the given conditions are a fortiori satisfied.
For comparison with Diophantus's solutions of quadratic
equations we may refeV to a few of his solutions of
(3) Simultaneous equations involving quadratics.
In I. 27, 28, and 30 we have the following pairs of equations.
(a) ^rj = 2a] (/3) £ + v = 2a\ (y) £- V = 2a)
I use the Greek letters for the numbers required to be found
as distinct from the one unknown which Diophantus uses, and
which I shall call x.
In (a), he says, let £ — r] = 2x (£ > rj).
It follows, by addition and subtraction, that £ = a + x,
tj — a — x\
therefore £rj — (a + x) (a — x) = a 2 — x 2 = B,
and x is found from the pure quadratic equation.
In (/3) similarly he assumes £ — t] = 2x, and the resulting
equation is £ 2 + rj 2 — (a + x) 2 + (a — x) 2 = 2 (a 2 + x 2 ) = B.
In (y) he puts £ + ?; = 2x and solves as in the case of (a).
(4) Cubic equation.
Only one very particular case occurs. In VI. 17 the problem
leads to the equation
x 2 + 2x + 3 = x* + 3x — 3x 2 — 1.
Diophantus says simply ' whence x is found to be 4 '. In fact
the equation reduces to
x z + x = 4x 2 + 4.
Diophantus no doubt detected, and divided out by, the common
factor x 2 + 1 , leaving x = 4.
1523.2 H ll
466 DIOPHANTUS OF ALEXANDRIA
(B) Indeterminate equations.
Diophantus says nothing of indeterminate equations of the
first degree. The reason is perhaps that it is a principle with
him to admit rational fractional as well as integral solutions,
whereas the whole point of indeterminate equations of the
first degree is to obtain a solution in integral numbers.
Without this limitation (foreign to Diophantus) such equa-
tions have no significance.
(a) Indeterminate equations of the second degree.
The form in which these equations occur is invariably this :
one or two (but never more) functions of x of the form
Ax 2 -f Bx + G or simpler forms are to be made rational square
numbers by finding a suitable value for x. That is, we have
to solve, in the most general case, one or two equations of the
form Ax 2 + Bx + C = y 2 .
(1) Single equation.
The solutions take different forms according to the particular
values of the coefficients. Special cases arise when one or
more of them vanish or they satisfy certain conditions.
1. When A or G or both vanish, the equation can always
be solved rationally.
Form Bx = y 2 .
Form Bx + G = y 2 .
Diophantus puts for y 2 any determinate square m 2 , and x is
immediately found.
Form Ax 2 + Bx — y 2 .
Diophantus puts for y any multiple of x, as — x.
2. The equation Ax 2 + C = y 2 can be rationally solved accord-
ing to Diophantus :
(a) when A is positive and a square, say a 2 ;
in this case we put a 2 x 2 + G = (ax ± m) 2 , whence
C — m 2
x= -\
(m and the sign being so chosen as to give x a positive value) ;
INDETERMINATE EQUATIONS 467
(ft) when C is positive and a square, say c 2 ;
in this case Diophantus puts Ax 2 + c 2 — (mx±c) 2 , and obtains
2mc
x = +
-A
m'
(y) When one solution is known, any number of other
solutions can be found. This is stated in the Lemma to
VI. 15. It would be true not only of the cases ±Ax 2 + C = y 2 ,
but of the general case Ax 2 + Bx + C = y 2 . Diophantus, how-
ever, only states it of the case Ax 2 — C — y 2 .
His method of finding other (greater) values of x satisfy-
ing the equation when one (x ) is known is as follows. If
A x 2 -r C = q 2 , he substitutes in the original equation (x + x)
for x and (q — kx) for y, where k is some integer.
Then, since A (x + x) 2 — C = (q — kx) 2 , while Ax 2 — C = q 2 ,
it follows by subtraction that
2x(Ax + kq) — x 2 (k 2 — A),
whence x — 2 (Ax + kq) / (k 2 — A),
and the new value of x is x -\ jo° a '
Form Ax 2 — c 2 = y 2 .
Diophantus says (VI. 14) that a rational solution of this
case is only possible when A is the sum of two squares.
[In fact, if x = p/q satisfies the equation, and Ax 2 — c 2 = k 2 t
we have Ap 2 = c 2 q 2 + k 2 q 2 ,
Form Ax 2 + C = y 2 .
Diophantus proves in the Lemma to VI. 12 that this equa-
tion has an infinite number of solutions when A + C is a square,
i. e. in the particular case where x = 1 is a solution. (He does
not, however, always bear this in mind, for in III. 10 he
regards the equation 52x 2 +12 = y 2 as impossible though
52 + 12 = 64 is a square, just as, in III. 11, 266a? 2 - 10 = y 2
is regarded as impossible.)
Suppose that A + C = q 2 ; the equation is then solved by
Hh 2
468 DIOPHANTUS OF ALEXANDRIA
substituting in the original equation 1 + x for x and (q — kx)
for y, where k is some integer.
3. Form Ax 2 + Bx + C=y 2 .
This can be reduced to the form in which the second term is
wanting by replacing x by z — —j •
2 XL
Diophantus, however, treats this case separately and less
fully. According to him, a rational solution of the equation
Ax' 1 + Bx +C = y 2 is only possible
(a) when A is positive and a square, say a 2 ;
(/?) when C is positive and a square, say c 2 ;
(y) when ^B 2 — AC is positive and a square.
In case (a) y is put equal to (ax — m), and in case (/3) y is put
equal to (mx — c).
Case (y) is not expressly enunciated, but occurs, as it
were, accidentally (IV. 31). The equation to be solved is
3 x + 1 8 — x 2 = y 2 . Diophantus first assumes 3 x + 1 8 — x 2 = 4 x 2 ,
which gives the quadratic 3^+18 = 5 sc 2 ; but this 'is not
rational '. Therefore the assumption of 4 x 2 for y 2 will not do,
' and we must find a square [to replace 4] such that 1 8 times
(this square + 1 ) 4- (f ) 2 may be a square '. The auxiliary
equation is therefore 18(m 2 + 1) + § = y 2 , or 7 2 m 2 + 81= a
square, and Diophantus assumes 72 m 2 + 8 1 = (8 m + 9) 2 , whence
m= 18. Then, assuming 3 x + 18 — x 2 = (1 8) 2 ^ 2 , he obtains the
equation 325 # 2 — 3x— 18 = 0, whence x — / 2 t> ^ na ^ i s > 2V
(2) Double equation.
The Greek term is SLTrXoicroT-qs, SlttXtj IcroT'qs or SlttXyj io-cdctis.
Two different functions of the unknown have to be made
simultaneously squares. The general case is to solve in
rational numbers the equations
mx 2 + oc x + a — u 2 j
nx 2 -\- fix+ b = w 2 )
The necessary preliminary condition is that each of the two
expressions can be made a square. This is always possible
when the first term (in x 2 ) is wanting. We take this simplest
case first.
INDETERMINATE EQUATIONS 469
1 . Double equation of the first degree.
The equations are
a x + a = u 2 ,
. ftx + b — w 2 .
Diophantus has one general method taking slightly different
forms according to the nature of the coefficients.
(a) First method of solution.
This depends upon the identity
{i(p+q)V'-{i(p-q)V- = pq-
If the difference between the two expressions in x can be
separated into two factors p, q, the expressions themselves
are equated to {i(p + q}} 2 and { \ (p — q) } 2 respectively. As
Diophantus himself says in II. 11, we ' equate either the square
of half the difference of the two factors to the lesser of the
expressions, or the square of half the sum to the greater',
We will consider the general case and investigate to what
particular classes of cases the method is applicable from
Diophantus's point of view, remembering that the final quad-
ratic in x must always reduce to a single equation.
Subtracting, we have (oc — ft) x + (a — b) = u 2 — w 2 .
Separate (oc — ft) x + (a — b) into the factors
p, {(a-ft)x + (a-b)} /p.
We write accordingly
(oi—ft)x + (a-b)
n + w — — -,
P
u + %v = p.
m, o i {(oc~ft)x+(a—b) ) 2
Thus u- = (xx + a = i \- — ' + p[ ;
t p s
therefore {(a- ft) X -\-a-b+p 2 } 2 = 4p 2 (ax + a).
This reduces to
(a-ft) 2 x 2 + 2x{(oc~ft)(u-b)-p 2 (oc + ft)}
+ [a - b) 2 - 2 p 2 (a + b)+ jA = 0.
470 DIOPHANTUS OF ALEXANDRIA
In order that this equation may reduce to a simple equation,
either
(1) the coefficient of x 2 must vanish, or oc — /? = 0,
or (2) the absolute term must vanish, that is,
£> 4 - 2 p 2 (a + b) + (a- b) 2 = 0,
or { p 2 — (a + b) } 2 = 4 ab t
so that ab must be a square number.
As regards condition (1) we observe that it is really sufficient
if ocn 2 = /3m 2 , since, if oc x -f a is a square, (otx + a) n 2 is equally
a square, and, if fix + b is a square, so is (ftx + tym 2 , and
vice versa.
That is, (1) we can solve any pair of equations of the form
ocm 2 x + a = w 2]
an 2 x + b — w 2
Multiply by n 2 , m 2 respectively, and we have to solve the
equations
ocm 2 n 2 x+an 2 = u' 2
oc m 2 n 2 x + bm 2 = w' 2 \
Separate the difference, an 2 — bm 2 , into two factors p, q and
put u f ± iv' = p,
u'+w' = q\
therefore u' 2 = \(p + q) 2 , w' 2 = \(p — q) 2 ,
and a m 2 n 2 x + an 2 = %(p + <?) 2 ,
a m 2 n 2 x + 6m/ 2 = i(p — q) 2 ',
and from either of these equations we get
i (p 2 + q 2 ) — ^ (an 2 + bm 2 )
x —
i £> nrt 2
since £>g = aw 2 — 6m 2 .
Any factors £>, <Z can be chosen provided that the resulting
value of x is 'positive.
INDETERMINATE EQUATIONS 471
Ex. from Diophantus :
65- 6x = u 2 ) /t ^ t
9 ; ( IV - 32 )
65-24x = w 2 J v '
therefore 200 - 24 x = u' 2
G5 — 2AX = i</ 2
The difference = 195 = 15. 13, say;
therefore J(15~ 13) 2 = 65-24^; that is, 24# = 64, and x = §.
Taking now the condition (2) that ab is a square, we see
that the equations can be solved in the cases where either
a and b are both squares, or the ratio of a to b is the ratio of
a square to a square. If the equations are
ocx + c 2 = u 2 ,
fix + d 2 = w 2 ,
and factors are taken of the difference between the expressions
as they stand, then, since one factor p, as we saw, satisfies the
equation { p 2 — (c 2 + d 2 ) } 2 = 4 c 2 d* t
we must have p = c ± d.
Ex. from Diophantus:
10^+9 = it 2 '
5x + 4 = w 2
The difference is 5#+5 = 5(#-f-l); the solution is given by
(±x+3) 2 = 10^ + 9, and x = 28.
Another method is to multiply the equations by squares
such that, when the expressions are subtracted, the absolute
term vanishes. The case can be worked out generally, thus.
Multiply by d 2 and c 2 respectively, and we have to solve
ocd 2 x-\-c 2 d 2 = u 2 1
pc 2 x + c 2 d 2 = w 2 \'
Difference = (ocd 2 — /3c 2 )x = px . q say.
Then x is found from the equation
ocd 2 x + c 2 d 2 = J (px + q) 2 ,
which gives p 2 x 2 -\- 2x(pq — 2ocd 2 ) + q 2 -4c 2 d 2 = 0,
(III. 15)
472 DIOPHANTUS OF ALEXANDRIA
or, since pq = (xd 2 — @c 2 ,
p 2 x 2 -2x((xd 2 + l3c 2 ) + q 2 -4e 2 d 2 = 0.
In order that this may reduce to a simple equation, as
Diophantus requires, the absolute term must vanish, so that
q = 2cd. The method therefore only gives one solution, since
q is restricted to the value 2cd.
Ex. from Diophantus :
Sx + 4 =u 2 ) /TTr
(IV. 39)
6^ + 4 = w 2 )
Difference .2 x; q necessarily taken to be 2 a/4 or 4; factors
therefore \x, 4. Therefore Sx + 4 = \ {\x + 4) 2 , and a; = 112.
(/?) Second method of solution of a double equation of the
first degree.
There is only one case of this in Diophantus, the equations
being of the form
hx + n 2 = u 2 )
(h+f)x + n 2 =w 2 )
Suppose hx + n 2 = (y + n) 2 ; therefore hx = y 2 + 2 ny,
•f
and (h +f ) x + n 2 — (y + n) 2 + r (y 2 + 2 ny).
It only remains to make the latter expression a square,
which is done by equating it to {jiy — nf.
The case in Diophantus is the same as that last mentioned
(IV. 39). Where I have used y, Diophantus as usual contrives
to use his one unknown a second time.
2. Double equations of the second degree.
The general form is
Ax 2 +Bx +C = u 2 \
A'x 2 + B'x + C'=iv 2 \ '
but only three types appear in Diophantus, namely
. . p 2 x 2 + (xx + a — u 2 ) . •
(1) J- , where, except in one case, a = 6.
p 2 x 2 + px + b — w l )
INDETERMINATE EQUATIONS 473
x 2 + <xx + a = u 2 1
' I3x 2 + a=w 2 l'
(The case where the absolute terms are in the ratio of a square
to a square reduces to this.)
In all examples of these cases the usual method of solution
applies.
x ocx 2 + ax = u 2 )
(3)
V ' px 2 +bx = w 2 )
The usual method does not here serve, and a special artifice
is required.
Diophantus assumes u 2 = m 2 x 2 .
Then x = a/(m 2 — a) and, by substitution in the second
equation, we have
B( — ) + — ? which must be made a square,
\m* — ol/ wZ — oc
or a 2 /3 + ba(m 2 — a) must be a square.
We have therefore to solve the equation
abm 2 + a(a/3 — ocb) = y 2 ,
which can or cannot be solved by Diophantus's methods
according to the nature of the coefficients. Thus it can be
solved if (a/3 — ab)/a is a square, or if a/b is a square.
Examples in VI. 12, 14.
(b) Indeterminate equations of a degree higher than the
second.
(1) Single equations.
There are two classes, namely those in which expressions
in x have to be made squares or cubes respectively. The
general form is therefore
Ax n + Bx n ~ l + ... +Kx-L = y 2 or y 3 .
In Diophantus n does not exceed 6, and in the second class
of cases, where the expression has to be made a cube, n does
not generally exceed 3.
474 DIOPHANTUS OF ALEXANDRIA
The species of the first class found in the Arithmetica are
as follows.
1 . Equation Ax 3 + Bx 2 + Gx + d 2 — y 2 .
As the absolute term is a square, we can assume
or we might assume y = m 2 x 2 + nx + d and determine m, n so
that the coefficients of x, x 2 in the resulting equation both
vanish.
Diophantus has only one case, x z — 3 x 2 + 3x + 1 = y' 2 (VI. 1 8),
and uses the first method.
2. Equation A # 4 -f Bx z + Ox 2 4- Dx .+ E — y 2 , where either A or
E is a square.
If A is a square ( = a 2 ), we may assume y = ax 2 H x + w,
determining ?i so that the term in x 2 in the resulting equa-
tion may vanish. If E is a square (= e 2 ), we may assume
y = m# 2 + —x + e, determining m so that the term in x 2 in the
resulting equation may vanish. We shall then, in either case,
obtain a simple equation in x.
3. Equation Ax^ + 0# 2 + E = y 2 , but in special cases only where
all the coefficients are squares.
4. Equation Ax* + E = y 2 .
The case occurring in Diophantus is a? 4 + 97 — y 2 (V. 29).
Diophantus trie's one assumption, y = x 2 — 1 0, and finds that
this gives x 2 = 2%, which leads to no rational result. He
therefore goes back and alters his assumptions so that he
is able to replace the refractory equation by x 4: + 337 = y 2 ,
and at the same time to find a suitable value for y, namely
y — x 2 — 25, which produces a rational result, x — --£-.
5. Equation of sixth degree in the special form
x 6 —Ax z + Bx + c 2 — y 2 .
Putting yz=zx z + c, we have — Ax 2 + B = 2cx 2 , and
B B
x 2 — —. , which gives a rational solution if — A — — - is
A + 2c . 5 A + 2c
INDETERMINATE EQUATIONS 475
a square. Where this does not hold (in IV. 18) Diophantus
harks back and replaces the equation x G — I6x ?j + 02 + 64 = y 2
by another, a? 6 — 128a; 3 + x + 4096 = y 2 .
Of expressions which have to be made cubes, we have the
following cases.
1. Ax 2 + Bx + C = y 3 .
There are only two cases of this. First, in VI. 1, 03 2 -— 4 a; + 4
has to be made a cube, being already a square. Diophantus
naturally makes «-2a cube.
Secondly, a peculiar case occurs in VI. 1 7, where a cube has
to be found exceeding a square by 2. Diophantus assumes
(x— l) 3 for the cube and (x + l) 2 for the square. This gives
a; 3 -3a; 2 +3a:-l = x 2 + 2x + 3,
or x"' + x — 4 a? 2 + 4. We divide out by x 2 +l, and a; = 4. It
seems evident that the assumptions were made with knowledge
and intention. That is, Diophantus knew of the solution 27
and 25 and deliberately led up to it. It is unlikely that he was
aware of the fact, observed by Fermat, that 27 and 25 are the
only integral numbers satisfying the condition.
2. Ax 3 + Bx 2 + Cx + D*= y 3 , where either A or D is a cube
number, or both are cube numbers. Where A is a cube (a 3 ),
we have only to assume y = ax+ —-r } , and where D is a cube
G
(d 3 ), y— —-j 9 x + d. Where A = a 3 and D = d 3 , we can use
o CL"
either assumption, or put y — ax + d. Apparently Diophantus
used the last assumption only in this case, for in IV. 27 he
rejects as impossible the equation 8x 3 — x 2 + 8x—l=y 3 ,
because the assumption y = 2x— 1 gives a negative value
x = — xx, whereas either of the above assumptions gives
a rational value.
(2) Double equations.
Here one expression has to be made a square and another
a cube. The cases are mostly very simple, e.g. (VI. 19)
4a; + 2 = y 3 )
2x + \ =Z 2 \'
thus y 3 — 2z 2 , and z = 2.
476 DIOPHANTUS OF ALEXANDRIA
More complicated is the case in VI. 21 :
2# 2 + 2# = y 2
x* + 2x 2 + x = z 3
Diophantus assumes y = mx, whence x = 2/(m 2 — 2), and
/ 2 y / 2 \ 2 2
W-~2/ + Vm 2 - 2/ + m^^2 ~ *''
2971
or 7 — 5 ^ = 2 3 -
(m 2 -2) 3
We have only to make 2??i 4 , or 2 m, a cube.
II. Method of Limits.
As Diophantus often has to find a series of numbers in
order of magnitude, and as he does not admit negative
solutions, it is often necessary for him to reject a solution
found in the usual course because it does not satisfy the
necessary conditions ; he is then obliged, in many cases, to
find solutions lying within certain limits in place of those
rejected. For example :
1. It is required to find a value of x such that some power of
it, x n , shall lie between two given numbers, say a and b.
Diophantus multiplies both a and b by 2 n , 3 n , and so on,
successively, until some nth power is seen which lies between
the two products. Suppose that c n lies between ap n and bp n ;
then we can put x = c/p, for (c / ' p) n lies between a and b.
Ex. To find a square between l£ and 2. Diophantus
multiplies by a square 64; this gives 80 and 128, between
which lies 100. Therefore (V 0- ) 2 or ff solves the problem
(IV. 31 (2)).
To find a sixth power between 8 and 16. The sixth powers
of 1, 2, 3, 4 are 1, 64, 729, 4096. Multiply 8- and 16 by 64
and we have 512 and 1024, between which 729 lies; - 7 g 2 4 9 - is
therefore a solution (VI. 21).
2. Sometimes a value of x has to be found which will give
METHOD OF LIMITS 477
some function of x a value intermediate between the values
of two other functions of x.
Ex. 1. In IV. 25 a value of x is required such that 8/(x 2 + x)
shall lie between x and x + 1 .
One part of the condition gives 8 > x 3 + x 2 . Diophantus
accordingly assumes 8 = (o; + -§-) 3 = x a -{-x 2 + ^x + Jy, which is
> x 3 + x 2 . Thus x + •§■ = 2 or 3? = ■§ satisfies one part of
the condition. Incidentally it satisfies the other, namely
8/(x 2 + x) < x+l. This is a piece of luck, and Diophantus
is satisfied with it, saying nothing more.
Ex. 2. We have seen how Diophantus concludes that, if
i(a;2_60) > x > !(.£ 2 -60),
then x is not less than 1 1 and not greater than 12 (V. 30).
The problem further requires that x 2 — 60 shall be a square.
Assuming a? 2 — 60 = (x— m) 2 , we find x = (m 2 + 60)/ 2 m.
Since x > 1 1 and < 1 2, says Diophantus, it follows that
24m > m 2 + 60 > 22 m;
from which he concludes that m lies between 19 and 21.
Putting m = 20, he finds x — \\\.
III. Method of approximation to Limits.
Here we have a very distinctive method called by Diophantus
wapKroTrjs or Trapio-oTrjTos dycoyrj. The object is to solve such
problems as that of finding two or three square numbers the
sum of which is a given number, while each of them either
approximates to one and the same number, or is subject to
limits which may be the same or different.
Two examples will best show the method.
Ex. 1. Divide 13 into two squares each of which > 6 (V. 9).
Take half of 13, i.e. 6J, and find what small fraction 1 /x 2
added to it will give a square ;
thus 6 J H — 5 j or 26 + — > must be a square.
x y
478 DIOPHANTUS OF ALEXANDRIA
Diophantus assumes
26+ ^=( 5 +^)' or 26t/ 2 +1 = (6y + lf,
whence
y = 10, and 1/t/ 2 = ft^ . i.e. l/x 2 = ^ft ; and 64 + ^fo = (ft) 2 .
[The assumption of 5 H — as the side is not haphazard : 5 is
chosen because it is the most suitable as giving the largest
rational value for y.']
We have now, says Diophantus, to divide 13 into two
squares each of which is as nearly as possible equal to (ft) 2 .
Now 13 = 3 2 + 2 2 [it is necessary that the original number
shall be capable of being expressed as the sum of two squares] ;
and 3 > ft by A,
while 2 < ft by ft.
But if we took 3— 5 %, 2+ ft as the sides of two squares,
their sum would be 2(ft) 2 = - 5 5 2 o°o 2 -> which is > 13.
Accordingly we assume 3 — 9#, 2 + Use as the sides of the
required squares (so that x is not exactly ■£§ but near it).
Thus (3-9#) 2 + (2 + lla;) 2 = 13,
and we find x = T f T .
The sides of the required squares are ffy, f of •
Ex. 2. Divide 10 into three squares each of which > 3
(V.ll).
[The original number, here 1 0, must of course be expressible
as the sum of three squares.]
Take one-third of 10, i.e. 3 J, and find what small fraction
\/x l added to it will make a square; i.e. we have to make
1 • 9 , . 1
3 ^"l — 2 a square, i.e. 30+ -j must be a square, or 30 H — g
xx y
= a square, where 3/x = l/y.
Diophantus assumes
30i/ 2 + l = (5y + l) 2 ,
the coefficient of y, i.e. 5, being so chosen as to make 1 /y as
small as possible ;
METHOD OF APPROXIMATION TO LIMITS 479
therefore y = 2, and 1 /x 2 = 3% ; and Sj + ^g = ±jg-, a square.
We have now, says Diophantus, to divide 10 into three
squares with sides as near as may be to -^.
Now 10 = 9 + 1 =3 2 + (f) 2 + (f) 2 .
Bringing 3, f- , f and ^ to a common denominator, we have
90 18 24 onrl 55
30 > 3~0> 3~0 d/11U 3 '
and 3 > || by §f ,
5 ^ 3 u J 30'
5" < "30 °y 30'
If now we took 3 — §J, f + |J , § + f§ as the sides of squares,
the sum of the squares would be 3 (V) 2 or " 3 3% 3- > which is > 10.
Accordingly we assume as the sides 3 — 35 #, § + 3 7 &•, f + 3 1 #,
where a? must therefore be not exactly 3V but near it.
Solving (3-35 l £) 2 + (! + 37£) 2 + (f + 31^) 2 = 10,
or 10-116^ + 3555^ 2 = 10,
we find x = ^A 5
thus the sides of the required squares are -VtiN ^ttt* VttS
the squares themselves are WA% 4 iS T5r£rr> VSfiflRtf-
Other instances of the application of the method will be
found in V. 10, 12, 13, 14.
Porisms and propositions in the Theory of Numbers.
I. Three propositions are quoted as occurring in the Porisms
(' We have it in the Porisms that ...'); and some other pro-
positions assumed without proof may very likely have come
from the same collection. The three propositions from the
Porisms are to the following effect.
1. If a is a given number and x, y numbers such that
x + a = m 2 , y + a = n 2 , then, if xy + a is also a square, m and n
differ by unity (V. 3).
[From the first two equations we obtain easily
xy + a = m 2 n 2 — a (m 2 + n 2 — 1) + a 2 ,
and this is obviously a square if m 2 + n 2 — 1 = 2 mn, or
m — n = ±1-]
480 DIOPHANTUS OF ALEXANDRIA
2. If 7Tb 2 , (m+ l) 2 be consecutive squares and a third, number
be taken equal to 2{m 2 + (m+ l) 2 } +2, or 4(m 2 + m+ 1), the
three numbers have the property that the product of any two
plus either the sum of those two or the remaining number
gives a square (V. 5).
[In fact, if X, Y, Z denote the numbers respectively,
XY+X+Y= (m 2 + m + 1) 2 5 XY + Z = (m 2 + m + 2)\
YZ+Y+Z = (2m 2 + 3m+3) 2 , YZ+X = (2m 2 + 3m + 2) 2 ,
ZX+Z+X = (2m 2 + m + 2) 2 , ZX + Y = (2m 2 + m + l) 2 .]
3. The difference of any two cubes is also the sum of two
cubes, i.e. can be transformed into the sum of two cubes
(V. 16).
[Diophantus merely states this without proving it or show-
ing how to make the transformation. The subject of the
transformation of sums and differences of cubes was investi-
gated by Vieta, Bachet and Fermat.]
II. Of the many other propositions assumed or implied by
Diophantus which are not referred to the Porisms we may
distinguish two classes.
1 . The first class are of two sorts ; some are more or less
of the nature of identical formulae, e.g. the facts that the
expressions {%(a + b)} 2 — ab and a 2 (a+ l) 2 + a 2 + (a+ l) 2 are
respectively squares, that a (a 2 — a) + a + (a 2 — a) is always a
cube, and that 8 times a triangular number plus 1 gives
a square, i.e. 8 ,\x (x+ 1) + 1 = (2x+ l) 2 . Others are of the
same kind as the first two propositions quoted from the
PorismSy e.g.
(1) If X z=a 2 x + 2a, Y ={a+\) 2 x+2(a+\) or, in other
words, if xX+ 1 = (ax+ l) 2 and xY "+ 1 = {(a+l)x+l } 2 ,
then XY +1 is a square (IV. 20). In fact
XY+l - {a(a + l)a? + (2a+l)} 2 .
(2) If X±a = m 2 , Y±a = (m+1) 2 , and Z= 2(X+F)-1,
then YZ±a, ZX±a, XY±a are all squares (V. 3, 4).
PORISMS AND PROPOSITIONS ASSUMED 481
In fact YZ±a = {(m+l)(2m+l) + 2a} 2 ,
ZX±a = {m(2m+ l) + 2a} 2 ,
XY±a= {.m(m+l) + a} 2 .
(3) If
X = m 2 + 2, F=(m+l)*+2, Z = 2{m 2 + (m + 1) 2 + 1 ] + 2,
then the six expressions
F^_(F+if), #X-(#+X), XY-(X+Y) }
YZ-X, ZX-Y, XY-Z
are all squares (V. 6).
In fact
YZ- (Y + Z) = {2m 1 + 3m + 3) 2 , F^-I=(2m 2 + 3m + 4) 2 ; &c.
2. The second class is much more important, consisting of
propositions in the Theory of Numbers which we find first
stated or assumed in the Arithmetica. It was in explana-
tion or extension of these that Fermat's most famous notes
were written. How far Diophantus possessed scientific proofs
of the theorems which he assumes must remain largely a
matter of speculation.
(a) Theorems on the composition of numbers as the sum
of two squares.
(1) Any square number can be resolved into two squares in
any number of ways (II. 8).
(2) Any number which is the sum of two squares can be
resolved into two other squares in any number of ways (II. 9).
(It is implied throughout that the squares may be fractional
as well as integral.)
(3) If there are two whole numbers each of which is the
sum of two squares, the product of the numbers can be
resolved into the sum of two squares in two ways.
In fact (a 2 + b 2 ) (c 2 + d 2 ) = (ac ± bd) 2 + {ad + be) 2 .
This proposition is used in III. 19, where the problem is
to find four rational right-angled triangles with the same
1523.2 I i
482 DIOPHANTUS OF ALEXANDRIA
hypotenuse. The method is this. Form two right-angled
triangles from (a, b) and (c, d) respectively, by which Dio-
phantus means, form the right-angled triangles
(a 2 + b 2 , a 2 -b 2 , 2ab) and (c 2 + d 2 , c 2 -d 2 , 2cd).
Multiply all the sides in each triangle by the hypotenuse of
the other; we have then two rational right-angled triangles
with the same hypotenuse (a 2 + b 2 ) (c 2 + d 2 ).
Two others are furnished by the formula above; for we
have only to ' form two right-angled triangles ' from (ac + bd,
ad — be) and from (ac — bd, ad + be) respectively. The method
fails if certain relations hold between a, b, c, d. They must
not be such that one number of either pair vanishes, i.e. such
that ad = be or ac = bd, or such that the numbers in either
pair are equal to one another, for then the triangles are
illusory.
In the case taken by Diophantus a 2 + b 2 = 2 2 + 1 2 = 5,
c 2 + d 2 = 3 2 + 2 2 = 1 3, and the four right-angled triangles are
(65, 52, 39), (65, 60, 25), (65, 63, 16) and (65, 56, 33).
On this proposition Fermat has a long and interesting note
as to the number of ways in which a prime number of the
form 4 n + 1 and its powers can be (a) the hypotenuse of
a rational right-angled triangle, (b) the sum of two squares.
He also extends theorem (3) above : ' If a prime number which
is the sum of two squares be multiplied by another prime
number which is also the sum of two squares, the product
will be the sum of two squares in two ways ; if the first prime
be multiplied by the square of the second, the product will be
the sum of two squares in three ways ; the product of the first
and the cube of the second will be the sum of two squares
in four ways, and so on ad infinitum'
Although the hypotenuses selected by Diophantus, 5 and 13,
are prime numbers of the form 4 n + 1 , it is unlikely that he
was aware that prime numbers of the form 4 n + 1 and
numbers arising from the multiplication of such numbers are
the only classes of numbers which are always the sum of two
squares ; this was first proved by Euler.
(4) More remarkable is a condition of possibility of solution
prefixed to V. 9, 'To divide 1 into two parts such that, if
NUMBERS AS THE SUMS OF SQUARES 483
a given number is added to either part, the result will be a
square.' The condition is in two parts. There is no doubt as
to the first, 'The given number must not be odd' [i.e. no
number of the form 4n + 3 or 4 w, — 1 can be the sum of two
squares] ; the text of the second part is corrupt, but the words
actually found in the text make it quite likely that corrections
made by Hankel and Tannery give the real meaning of the
original, ' nor must the double of the given number plus 1 be
measured by any prime number which is less by 1 than a
multiple of 4 '. This is tolerably near the true condition
stated by Fermat, ' The given number must not be odd, and
the double of it increased by 1 , when divided by the greatest
square which measures it, must not be divisible by a prime
number of the form 4 n — 1 .'
(ft) On numbers which are the sum of three squares.
In V. 11 the number 3<x+l has to be divisible into three
squares. Diophantus says that a 'must not be 2 or any
multiple of 8 increased by 2 '. That is, ' a number of the
form 24n + 7 cannot be the sum of three squares '. As a matter
of fact, the factor 3 in the 24 is irrelevant here, and Diophantus
might have said that a number of the form 8^ + 7 cannot be
the sum of three squares. The latter condition is true, but
does not include all the numbers which cannot be the sum of
three squares. Fermat gives the conditions to which a must be
subject, proving that 3<x+ 1 cannot be of the form 4 n (24&H- 7)
or 4 n (8k+ 7), where Jc = or any integer.
(y) Composition of numbers as the sum of four squares.
There are three problems, IV. 29, 30 and V. 14, in which it
is required to divide a number into four squares. Diophantus
states no necessary condition in this case, as he does when
it is a question of dividing a number into three or two squares.
Now every number is either a square Vr the sum of two, three
or four squares (a theorem enunciated by Fermat and proved
by Lagrange who followed up results obtained by Euler), and
this shows that any number can be divided into four squares
(admitting fractional as well as integral squares), since any
square number can be divided into two other squares, integral
ii 2
484 DIOPHANTUS OF ALEXANDRIA
or fractional. It is possible, therefore, that Diophantus was
empirically aware of the truth of the theorem of Fermat, but
we cannot be sure of this.
Conspectus of the Arithmetical with typical solutions.
There seems to be no means of conveying an idea of the
extent of the problems solved by Diophantus except by giving
a conspectus of the whole of the six Books. Fortunately this
can be done by the help of modern notation without occupying
too many pages.
It will be best to classify the propositions according to their
character rather than to give them in Diophantus's order. It
should be premised that x, y, z . . . indicating the first, second
and third . . . numbers required do not mean that Diophantus
indicates any of them by his unknown (9) ; he gives his un-
known in each case the signification which is most convenient,
his object being to express all his required numbers at once in
terms of the one unknown (where possible), thereby avoiding the
necessity for eliminations. Where I have occasion to specify
Diophantus's unknown, I shall as a rule call it £, except when
a problem includes a subsidiary problem and it is convenient
to use different letters for the unknown in the original and
subsidiary problems respectively, in order to mark clearly the
distinction between them. When in the equations expressions
are said to be = u 2 , v 2 , w 2 , t 2 ... this means simply that they
are to be made squares. Given numbers will be indicated by
a, b, c ... on, n ... and will take the place of the numbers used
by Diophantus, which are always specific numbers.
Where the solutions, or particular devices employed, are
specially ingenious or interesting, the methods of solution will
be shortly indicated. The character of the book will be best
appreciated by means of such illustrations.
[The problems marked with an asterisk are probably
spurious.]
(i) Equations of the first degree with one unknown.
I. 7. x — a — m(x — b).
I. 8. x + a = m (x + b).
DETERMINATE EQUATIONS 485
I. 9. a — x = m(b — x).
I. 10. x + b = m{a — x).
I. 11. x + b = m(x — a).
1.39. (a + x)b+(b + x)a = 2(a + b)x, \
or (a + b) x + (b + x)a = 2 (a + a?) b, L (a > />)
or (a + b)x + (a + x)b = 2 (b + x)a.)
Diophantus states this problem in this form, ' Given
two numbers (a, b), to find a third number (x) such that
the numbers
(a + x)b, (b + x)a, (a + b)x
are in arithmetical progression.'
The result is of course different according to the order
of magnitude of the three expressions. If a > b (5 and 3
are the numbers in Diophantus), then (a-\-x)b < (b + x)a;
there are consequently three alternatives, since (a + x)b
must be either the least or the middle, and (b + x) a either
the middle or the greatest of the three products. We may
have
(a + x) b < (a + b) x < (b + x)a,
or (a + b) x < {a + x) b < (b + x) a,
or (a + x) b < (b + x) a < (a + b) x,
and the corresponding equations are as set out above.
(ii) Determinate systems of equations of the first degree.
I. 1. x + y = a, x — y = b.
(I. 2. x + y = a, x = my,
ll. 4. x — y = a, x = my.
I. 3. x + y = a, x = my + b.
/ 11.
j I. 5. x + y = a, — x + - y — b, subject to necessary condition.
I. 6. x + y — a, — x y = b,
V lib lb
>)
y (x 1 >x 2 , y 1 >y 2 , z 1 >z 2 ).
486 DIOPHANTUS OF ALEXANDRIA
I. 12. x 1 + x 2 = y 1 + y 2 = a ) x 1 = my 2 ,y 1 ==nx 2 (oB 1 >x 2 ,y 1 >y 2 ).
I. 1 3. x x + x 2 = y 1 + y 2 — z t + z 2 — a'
X l = ™#2 1 Vl = nZ 2 1 -1 = 2^2 J
»
I. 15. flj + a = m(2/ — a), y -\-b — n(x—b).
[Diophantus puts y = £ + a, where £ is his unknown.]
r I. 16. y + z = a, z + x=zb, x + y = c. [Dioph. puts £=x + y + z.]
I. 17. y + s + w = a, z + iv + x = b,w + x + y = c,x + y + z = d.
[x + y + z + w = g.]
I. 18. y + z — x = a, z + x — y = b, x + y — z = c.
[Dioph. puts 2 £ = x + y + z.]
I. 19. 2/ + + W — # = a, z + w + a? — y = b, w + x + y — z=c,
x + y + z — w = d.
[2£ = x + y + z + w.]
I. 20. a? + 2/ r£ «, = a, x + y = mz, y + z = nx.
I. 2 1 . x = y + — z, y = z H — ic, = a + - ty (where «>w>2),
with necessary condition.
11.18*. x-Q^x + a) + (£* + *) = 2/- (^ + 6)+ ^ + a)
= ~C>* + C ) + G;2/ + O 5 * ; + 2/ + =
a.
[Solution wanting.]
(iii) Determinate systems of equations reducible to the
first degree.
I. 26. ax — a 2 , bx = oc.
I. 29. x + y = a, x 2 — y 2 = b. [Dioph. puts 2£ = x — y.']
{I. 31. a? = my, x 2 + y 2 = w(a? + 2/).
I. 32. a? = m?/, x 2 + y 2 = n(x — y).
I. 33. a? == m?/, x 2 — y 2 = n(x + y).
I. 34. a; = 7>t2/, x 2 —y 2 = n(x — y).
I. 34. Cor. 1. a? = m^/, 033/ = n(x + y).
Cor. 2. a? = m^/, «?/ = n{x~y).
DETERMINATE EQUATIONS 487
(I. 35. x = my, y 2 = nx.
1 1. 36. x = my, y 2 = ny.
I. 37. x = my, y 2 — n(x + y).
I. 38. x = my, y 2 = n(x — y).
I. 38. Cor. x = my, x 2 = ny.
„ x — my, x 2 = nx.
' „ x = my, x 2 = n{x + y).
„ x = my, x 2 = n(x — y).
II. 6*. x — y = a, x 2 — y 2 = x — y + b.
IV. 36. yz — m(y + z), zx = n(z + x), xy = p(x + y).
[Solved by means of Lemma : see under (vi) Inde-
terminate equations of the first degree.]
(iv) Determinate systems reducible to equations of
second degree.
I. 27. x + y = a, xy = b.
[Dioph. states the necessary condition, namely that
J a 2 — b must be a square, with the words eari Se tovto
irXaaiiaTLKov, which no doubt means 'this is of the
nature of a formula (easily obtained)'. He puts
*-v = 2 £]
I. 30. x — y — a, xy = b.
[Necessary condition (with the same words) 4 b + a 2 =
a square, x + y is put = 2 £.]
I. 28. x + y = a, x 2 + y 2 = b.
[Necessary condition 2 b — a 2 = a square, x — y = 2 £.]
(TV. 1. x* + y z = a, x + y = b.
[Dioph. puts x — y— 2^, whence x = \b + £, y = \b — £.
The numbers a, b are so chosen that (a — j6 3 )/3& is
a square.]
IV. 2. x 3 - y'* = a, x — y — b.
[x + y = 21]
488 DIOPHANTUS OF ALEXANDRIA
IV. 1 5. (y + z) x = a, (z + x) y = b, (x + y)z — c.
[Dioph. takes the third number z as his unknown
thus x + y = c/z.
Assume x = p/z, y = q/z. Then
pq
Z 2 +P = a >
-3+? = k
These equations are inconsistent unless p — q = a — b.
We have therefore to determine p } q by dividing c into
two parts such that their difference = a — b (cf. I. 1).
A very interesting use of the ' false hypothesis '
(Diophantus first takes two arbitrary numbers for p, q
such that p + q = c, and finds that the values taken have
to be corrected).
The final equation being —L +p — (l} where p, q are
z
determined in the way described, z 2 = pq/{& — r p) or
f pq/ (b — q), and the numbers a, b, c have to be such that
either of these expressions gives a square.]
IV. 34. yz + (y + z) = a 2 — 1, zx + (z + x) = b 2 - 1,
xy -f (x + y) = c 2 — 1 .
[Dioph. states as the necessary condition for a rational
solution that each of the three constants to which the
three expressions are to be equal must be some square
diminished by 1. The true condition is seen in our
notation by transforming the equations yz 4- (y + z) = (X,
zx + (z + x) — /3, xy + (x + y) = y into
(y+i)(z+i) = oc + i,
(z+l)(x+l) = £+1,
(x+\)(y+\) = y+1,
DETERMINATE EQUATIONS 489
whence «,+ != \ f + *> <* + i) \ &C. •
and it is only necessary that (a+ 1) (£+ 1) (y + 1) should
be a square, not that eac& of the expressions a + 1, £+ 1,
y + 1 should be a square.
Dioph. finds in a Lemma (see under (vi) below) a solu-
tion kv dopi(TT<p (indeterminately) of xy + (x + y) — k,
which practically means finding y in terms of &.]
IV. 35. yz-(y + z) = a 2 -\, zx-(z + x) = b 2 -l,
xy—(x + y) — c 2 — 1.
[The remarks on the last proposition apply mutatis
mutandis. The lemma in this case is the indeterminate
solution of xy — (x-\-y) = &.]
IV. 37. yz = a(x + y + z), zx = b(x-\-y + z), xy = c(x + y + z).
[Another interesting case of ' false hypothesis '. Dioph.
first gives x + y + z an arbitrary value, then finds that
the result is not rational, and proceeds to solve the new
problem of finding a value of x + y + z to take the place of
the first value.
i If w = x + y +z, we have x = cw/y, z = aw/y, so that
zx — acw 2 /y 2 = bw by hypothesis ; therefore y 2 = -j-w.
For a rational solution this last expression must be
a square. Suppose, therefore, that w = -y | 2 , and we have
x + y + z=™? 9 V = j-i> * = «£, tfj;=e£.
Eliminating x, y, z, we obtain g = (bc + ca + ab)/ ac,
and
& = (be + c<x + ab)/a, y = (bc + ca + ab)/b,
z — (bc + ca + ab)/c]
Lemma to V. 8. yz = a 2 } zx = b 2 , xy = c 2 .
490
DIOPHANTUS OF ALEXANDRIA
(v) Systems of equations apparently indeterminate but
really reduced, by arbitrary assumptions, to deter-
minate equations of the first degree.
I. 14. xy = m (x + y). [Value of y arbitrarily assumed.]
II. 3*. xy = m(x + y) } and xy = m(x — y).
*
II. 1*. (cf. I. 31). x 2 + y 2 = m(x + y).
\
■I II. 2*. (cf. I. 34). x 2 — y 2 = m(x — y). > [x assumed = 2 ?/.]
i
II. 4*. (cf. I. 32). x 2 -\-y 2 =m(x-y).
II. 5*. (cf. I. 33). x 2 -y 2 =m(x + y). t
II. 7*. x 2 — y 2 = m(x — y) +a. [Diopb. assumes x — y= 2.]
/ T ^ 1 1 11 11
I. 22. x a?+ -z = y y -\ x = z— - z+ ~y.
m p n m _p 7i
[Value of y assumed.]
i 1.23. a; a? H — w =. y y H a5 = # H — y
m q n m p n
= ty WH — 0. [Value of */ assumed.]
q p L ° J
1
1
1
1.24. sc + — (y + z) = y+-(z + x) = z+-(x + y).
[Value of y + z assumed.]
1.25. x + — (y + z + ty) = y + -(z + w + x)
= z + - (10 + aj + y) = w + - (x + y +z).
p\ at q \ u
[Value of y + + w assumed.]
11.17*. (cf. I. 22). x-(--x+a\ + (~z + c\
[Ratio of 35 to 2/ assumed.]
INDETERMINATE ANALYSIS 491
IV. 33. x + -y = m (y — -y)> y + -x = n(x — -xy
[ Solutions kv dopio-TG).
y practically found
[Dioph. assumes y = 1.]
(vi) Indeterminate equations of the first degree.
Lemma to IV. 34. xy + (x-\-y) = a.
., „ IV. 35. xy—(x + y) = <x.
„ IV. 36. xy = m(x + y). ) in terms of x -]
(vii) Indeterminate analysis of the second degree.
II. 8. x 2 + y 2 = a 2 .
[y 2 = a 2 — x 2 must be a square = (rnx — a) 2 , say.]
II. 9. x 2 + y 2 = a 2 + b 2 . [Put? ! x = g + a, y = mg — b.]
II. 10. x 2 — y 2 = a.
[Put a? = y + m, choosing m such that m 2 < a.]
II. 11. x + a = u 2 , x + b = v 2 .
II. 12. a — x = u 2 } b — x—v 2 .
III. 13. x — a = u 2 , x — b=v 2 .
[Dioph. solves II. 11 and 13, (1) by means of the
' double equation ' (see p. 469 above), (2) without a double
equation by putting x — £ 2 ±a and equating (£ 2 ±a) ±b
to (£ — m) 2 . In II. 12 he puts x = a — £ 2 .]
II. 14 = III. 21. x-\-y — a, x + z 2 = u 2 , 2/ + 2 2 = v 2 .
[Diophantus takes z as the unknown, and puts
u 2 = (0 + m) 2 , v 2 = (z + n) 2 . Therefore x = 2mz + m 2 ,
y = 2nz + n 2 , and z is found, by substitution in the first
equation, to be j ^ • In order that the solution
2 (m + 7i)
may be rational, m, n must satisfy a certain condition.
Dioph. takes them such that m 2 + n 2 < a, but it is suffi-
cient, if m > n, that a + m?i should be > n 2 .]
II. 15 = III. 20. x + y = a, z 2 — x = u 2 , z 2 — y = v 2 .
[The solution is similar, and a similar remark applies
to Diophantus's implied condition.]
492 DIOPHANTUS OF ALEXANDRIA
II. 1 6. x = my, a 2 + x = u 2 , a 2 + y = v 2 .
II. 19. x 2 -y 2 = m(y 2 -z 2 ).
II. 20. x 2 + y = u 2 , y 2 + x = v 2 .
[Assume y = 2fmx + m 2 , and one condition is satisfied.]
II. 21. x 2 — y = u 2 , y 2 — x — v 2 .
[Assume x = £ + m, y = 2 m £ + m 2 , *and one condition
is satisfied.]
ill. 22. o? 2 + (a? + y) = u 2 , 2/ 2 + (a; + 2/) = v 2 .
J [Put a; + 2/ = 2mx + m 2 .'\
Jl. 23. o? 2 — (a; + 2/) = u 2 , y 2 — (x + y) = v 2 .
II 24. (o3 + 2/) 2 + ^ = ^ (x+y) 2 +y = v 2 .
[Assume x — (m 2 — 1)£ 2 , y — {n 2 — 1)£ 2 , 02 + 3/ = £.]
II. 25. (a3 + 2/) 2 — x = u 2 , (x + y) 2 — y = v 2 .
II. 26. xy + x = u 2 , xy + y = v 2 , u + v = a.
[Put y = m 2 x — 1.]
II. 27. xy — x = u 2 , xy — y = v 2 , u + v = a.
II. 28. a; 2 2/ 2 + a; 2 = u 2 , x 2 y 2 + y 2 = ^ 2 .
JL 29. x 2 y 2 — x 2 = it 2 , x 2 y 2 — y 2 = v 2 .
II. 30. xy + (x + y) = u 2 , xy — (x + y) — v 2 .
[Since m 2 + w, 2 + 2 mw is a square, assume
xy = (m 2 + n 2 )£ 2 and x + y = 2mwf;
put a; = £>£, y = qg, where _pg = m 2 + ^ 2 ; then
(£> + #)£ = 2mn£ 2 .]
II. 31. xy + (x + y) = u 2 , xy — (x + y) = v 2 , x + y = w 2 .
[Suppose w 2 — 2.2m. m, which is a square, and use
formula (2 m) 2 + m 2 ±2.2m.m = a square.]
'II. 32. 2/ 2 + 2 = u 2 , s 2 + os = v 2 , x 2 + y = w 2 .
[V = I * = (2a| + a 2 ), a; = 2&(2a£ + a 2 ) + 6 2 .]
II. 33. y 2 — z — U 2 , z 2 — x = v 2 , x 2 — y — w 2 .
INDETERMINATE ANALYSIS 493
' II. 34. x 2 + (x + y + z) = u 2 , y 2 + (x + y + z) = v 2 ,
z 2 + (x + y + z) = w 2 .
[Since {^(m — n)} 2 + mn is a square, take any number
separable into two factors (on, n) in three ways. This
gives three values, say, p, q, r for -|(m — n). Put
x ■= p£> y = q£, z = r£, and x + y + z = mng 2 ; therefore
(P + Q + r )i == mng 2 , and £ is found.]
II. 35. x 2 — (x + y + z) = u 2 , y 2 — (x + y + z) = v 2 ,
v z 2 — (x + y + z) = w 2 .
[Use the formula { \ (on + W-) } 2 — mn — a square and
proceed similarly.]
III. 1*. (x + y + s) — x 2 = u 2 , (x + y + z) — y 2 = v 2 ,
(x + y + z)—z 2 = iv 2 .
'III. 2*. (# + y + s) 2 + x = it 2 , (x + y + z) 2 + y =. v 2 ,
(x + y + z) 2 + z = %v 2 .
III. 3*. (x + y + z) 2 -x = u 2 , (x + y + z) 2 -y = v 2 ,
(x + y + z) 2 — z = iv 2 .
III. 4*. x- (x + y + zf = u 2 , y-(x + y + zf = v 2 ,
— (a; + y + z) 2 = iv 2 .
III. 5. x + y + z — t 2 , y + z — x = u 2 , z + x — y = v 2 ,
x + y — z — iv 2 .
[The first solution of this problem assumes
t 2 = x + y + z = (i + 1) 2 , w 2 = 1, u 2 = i 2 ,
whence x, y, z are found in terms of £, and z + x — y
is then made a square.
The alternative solution, however, is much more ele-
gant, and can be generalized thus.
We have to find x, y, z so that
— x + y + z = a square
x — y + z = a square
x + y — z — a square
x + y + z = a square
Equate the first three expressions to a 2 , b 2 , c 2 , being
squares such that their sum is also a square = k 2 , say.
494 DIOPHANTUS OF ALEXANDRIA
Then, since the sum of the first three expressions is
itself equal to x + y + z, we have
x = i(6f + c 2 ), y = i(c 2 + a 2 ), z = i(« 2 + & 2 ).]
III. 6. x + y + z = t 2 , y + z = u 2 , z + x = v 2 , x + y — w 2 .
III. 7. x — y — y — z, y + z — u 2 , z + x = v 2 , x + y = w 2 .
| III. 8. x + 2/ + z + <x = t 2 , y + z + a = u 2 , z + x+a = v 2 ,
x + y + a — w 2 .
I III. 9. x + ?/ + z — <x = t 2 , y + z — a — u 2 , z + x — a = v 2 ,
\ x + y — a = w 2 .
III. 10. yz + a = u 2 , 2# + a = v 2 , xy + a = w 2 .
[Suppose yz + a = m 2 , and let y = (m 2 — a)£, z = 1 /£:
also let zx + a = ti 2 ; therefore a; = (w 2 — a)£.
We have therefore to make
(m 2 — a) (n 2 — a) g 2 + a a square.
Diophantus takes m 2 =25, a = 12, w, 2 = 16, and
arrives at 52£ 2 +12, which is to be made a square.
Although 52 . 1 2 + 12 is a square, and it follows that any
number of other solutions giving a square are possible
by substituting 1+rj for g in the expression, and so on,
Diophantus says that the equation could easily be solved
if 52 was a square, and proceeds to solve the problem of
finding two squares such that each increased by 12 will
give a square, in which case their product also will be
a square. In other words, we have to find m 2 and n 2
such that m 2 — a, n 2 — a are both squares, which, as he
says, is easy. We have to find two pairs of squares
differing by a. If
a = pq=p'q', {% (p-q)}* + a = {i(p + q)}\
and {h'(p'-q')} 2 +a = {Hp+q')} 2 ;
let, then, m 2 = { \ (p + q) } 2 , n 2 = { \ (p' + q') } 2 .]
III. 11. yz — a — u 2 , zx — a = v 2 , xy — a = w 2 .
£The solution is like that of III. 1 0. mutatis mutandis.]
fill. 12. yz + x = u 2 , zx + y = v 2 , xy + z = iv 2 .
(III. 13. yz — x— u 2 , zx'—y = v 2 , xy — z — tv 2 .
INDETERMINATE ANALYSIS 495
III. 14. yz + x 2 = u 2 , 2x + y 2 = v 2 , xy + z 2 — w 2 .
III. 15. yz+(y + z) = u 2 , zx + (z + x) = v 2 , xy + (x + y) = w 2 .
[Lemma. If a, a+l be two consecutive numbers,
a 2 (a + l) 2 + a 2 + (a + I) 2 is a square. Let
7/ = m 2 , z = (m + 1 ) 2 .
Therefore (m 2 + 2 m + 2) a; + (m + 1 ) 2 '
and (m 2 +l)o? + m 2
have to be made squares. This is solved as a double-
equation ; in Diophantus's problem m = 2.
Second solution. Let x be the first number, m the
second; then (m+l)x + m is a square = n 2 , say; there-
fore x = (n 2 — m)/(m+ 1), while y = m. We have then
(m + 1) + m = a square
/n 2 + l\ n 2 — m
and ( — - ) z + — — = a square
\m + 1 / -m-H 1
Diophantus has m = 3, n = 5, so that the expressions
to be made squares are with him
42 + 3 \
6iz + 5i\
This is not possible because, of the corresponding coeffi-
cients, neither pair are in the ratio of squares. In order to
substitute, for 6 J, 4, coefficients which are in the ratio
of a square to a square he then finds two numbers, say,
p } q to replace 5^, 3 such that pq+p + q = a square, and
(p + 1 ) / (q + 1 ) = a square. He assumes £ and 4 £ + 3 ,
which satisfies the second condition, and then solves for g }
which must satisfy
4 £ 2 + 8 £ + 3 = a square = (2£ -3) 2 , say,
which gives £ = t 3 q, 4£ + 3 — 4-|.
He then solves, for z, the third number, the double-
equation
5~z + 4-§- = square]
To^ + T 3 o = square]
496 DIOPHANTUS OF ALEXANDRIA
after multiplying by 25 and 100 respectively, making
expressions
130^+105)
130.T+ 30J
In the above equations we should only have to make
n 2 + 1 a square, and then multiply the first by n 2 + 1 and
the second by (m + l) 2 .
Diophantus, with his notation, was hardly in a position
to solve, as we should, by writing
(y + i)(z + i)=a 2 +l,
(Z + l)( X +l) = b 2 +l,
(x+l)(y+l) = c 2 +l,
which gives x + 1 = V { (b 2 + 1) (c 2 + I) /(a 2 + 1) }, &c]
III. 16. yz — (y + z) = u 2 , zx—(z + x) = v 2 , xy—{x + y) = w 2 .
[The method is the same mutatis mutandis as the
second of the above solutions.]
rill. 1 7. xy + (x + y) = u 2 , xy + x = v 2 , xy + y = w 2 .
llll. 18. xy — (x + y)= u 2 , xy — x = v 2 , xy — y — iv 2 .
(t 2
III. 1 9. (x x + x 2 + x. 6 + x^) 2 ± x x = j
I It/
^&'i -f- Xa ~r «£•> ~r "-'4/ jt *^s "~ 1
IT
'2
^t^j T ^'2 ' ^3 "■ ^4) "" i *^4 —
z</ 2
[Diophantus finds, in the way we have seen (p. 482),
four different rational right-angled triangles with the
same hypotenuse, namely (65, 52, 39), (65, 60, 25), (65,
56, 33), (65, 63, 16), or, what is the same thing, a square
which is divisible into two squares in four different ways ;
this will solve the problem, since, if h, p, b be the three
sides of a right-angled triangle, h 2 ±2pb are both squares.
INDETERMINATE ANALYSIS 497
Put therefore x x + x 2 + x 3 + x± = 65 £.
and x 1 = 2. 39. 52 £ 2 , ;r 2 = 2 . 25.60£ 2 , cc 3 = 2. 33. 56 £ 2 ,
x i = 2.16.63£ 2 ;
this gives 12768f = 65£, and £ = T ^ ¥ -]
(IV. 4. x 2 + y = u 2 , x + y = u.
tlV. 5. £c 2 + 2/ = Uj x + y = u 2 .
IV. 13. $ + 1 = t 2 , y+1 = u 2 , x + y + 1 = v 2 , 2/ — a; + 1 = to 2 ,
[Put a; = (mg + l) 2 — 1 = m 2 £ 2 + 2m^; the second and
third conditions require us to find two squares with x as
difference. The difference m 2 £ 2 + 2m g is separated into
the factors m 2 £ + 2m, £; the square of half the differ-
ence = {J(m, 2 — l)£ + m} 2 . Put this equal to y+1, so
that y = i(m 2 — l) 2 £ 2 + m(m 2 — 1) £ + m 2 — 1, and the
first three conditions are satisfied. The fourth gives
J (m 4 — 6 m 2 4- 1 ) £ 2 + (m 3 — 3 m) | + m 2 = a square, which
we can equate to (n£ — m) 2 .]
IV. 14. ^ 2 + 2/ 2 + z 2 = (^ 2 - 2/ 2 ) + (y 2 - z 2 ) + (x 2 ~z 2 ). (x>y> z)
IV. 16. x + 2/ 4- z — t 2 , x 2 + 2/ = '^ 2 > y 2 + z = v 2 , z 2 + x = %v 2 .
[Put 4m£ for 7/, and by means of the factors 2m£, 2
we can satisfy the second condition by making x equal
to half the difference, or m£ — 1. The third condition
is satisfied by subtracting (4m£) 2 from some square, say
(4m£+l) 2 ; therefore z = 8mg+l. By the first con-
dition 13m£ must be a square. Let it be 169 77 s ; the
numbers are therefore 13?? 2 — 1, 52?7 2 , 104?7 2 -f-l, and
the last condition gives 10816 ?; 4 + 221 ?? 2 = a square,
i.e. 10816t7 2 + 221 = a square = (104?? + l) 2 , say. This
gives the value of 77, and solves the problem.]
IV. 17. x + y + z — t 2 , x 2 — y — u 2 , y 2 — z — v 2 , z 2 — x = w 2 .
IV. 19. yz+1 = u 2 , zx + 1 = v 2 , xy+l = iv 2 .
[We are asked to solve this indeterminately {kv tco
dopi<TTG>). Put for yz some square minus 1, say m 2 £ 2
+ 2m£; one condition is now satisfied. Put z = £, so
thatj^/ = wi 2 £ + 2 m.
1523.2 K k
498 DIOPHANTUS OF ALEXANDRIA
Similarly we satisfy the second condition by assuming
zx — n 2 £ 2 + 2ng ; therefore x = n 2 g + 2 n. To satisfy the
third condition, we must have
(m 2 n 2 £ 2 + 2mn . m + n£ + 4m%) + 1 a square.
We must therefore have 4 mn + 1 a square and also
mn(m + n) = mn V(4mn-\- 1). The first condition is
satisfied by n = m-\- 1 , which incidentally satisfies the
second condition also. We put therefore yz = (m£ + l) 2 — 1
and zx— { (m + l)£ + 1 } 2 — 1, and assume that z = g, so that
y = m 2 g + 2m, x = (m + 1) 2 £ + 2(m + 1), and we have
shown that the third condition is also satisfied. Thus we
have a solution in terms of the undetermined unknown £.
The above is only slightly generalized from Diophantus.]
IV. 20. x 2 x z -\- 1 = r 2 , x z x Y -\~ 1 = s 2 , x 1 x 2 + 1 = t 2 ,
X-i X/i ~f~ 1 — iXi , Xc) Xi ~t~ J- — U , Xr, Xa ~t" i — w .
[This proposition depends on the last, x lt x 2 , x z being
determined as in that proposition. If x z corresponds to z
in that proposition, we satisfy the condition x 3 x 4 +l = w 2
by putting x 3 x± = {(m + 2)£ + 1 } 2 — 1, and so find x 4 in
terms of £, after which we have only two conditions more
to satisfy. The condition x 1 x 4: + 1 = square is auto-
matically satisfied, since
f (m + 1) 2 £ + 2 (m + 1)} { (m + 2) 2 £ + 2 (m + 2) } + 1
is a square, and it only remains to satisfy x 2 x i +l = square.
That is,
(m 2 |+2m) {(m + 2) 2 £ + 2(m + 2)} + 1
= m 2 (m+2) 2 f + 2m(m + 2)(2m + 2)^ + 4m(m+2) + 1
has to be made a square, which is easy, since the coefficient
of £ 2 is a square.
With Diophantus m = 1, so that x x = 4£ + 4, x 2 = £ + 2,
# 3 = £, # 4 = 9£ + 6, and 9£ 2 + 24£+13 has to be made
a square. He equates this to (3£— 4) 2 , giving £ = y 1 ^.]
IV. 21. xz = y 2 , x — y = u 2 , x — z = v 2 , y — z — w 2 . (x>y>z)
IV. 22. xyz + x = u 2 , xyz + y — v 2 , xyz + z = w 2 .
IV. 23. xyz — x = u 2 , xyz — y = i; 2 , xyz — z = iv 2 .
INDETERMINATE ANALYSIS 499
IV. 29. x 2 + y 2 + z 2 + iv 2 + x + y + z + w = a.
[Since x 2 + x+% is a square,
(x 2 + x) + (y 2 + y) + (z 2 + z) + (w 2 + iv) + 1
is the sum of four squares, and we only have to separate
a + 1 into four squares.]
I IV. 30. x 2 + y 2 + z 2 + iu 2 — (x + y + z + w) = a.
IV. 31. x + y — 1, (# + a) (# + &) = w 2 .
IV. 32. ^ + 2/4-^ = 0-, xy + z = u 2 , xy — z = v 2 .
IV. 39. x — y = m(y — z), y + z = u 2 , z + x = v 2 , x + y = to 2 .
IV. 40. x 2 — y 2 = m(y — z), y + z = u 2 , z + x = v 2 , x + y = w 2 .
V. 1. xz = y 2 , x — a = u 2 , y — a = v 2 , z — a = w 2 .
V. 2. xz — y 2 , x + a =u 2 , y + a = v 2 , z+a — w 2 .
( V. 3. x + a = r 2 , y + a = s 2 , z + a = £ 2 ,
2/0 + (X = u 2 , ;sa? + a = v 2 , xy + a = iy 2 .
V. 4. a? — a = r 2 , y — a = s 2 , z — a — t 2 ,
yz — a— u 2 , zx — a=v 2 , xy — a = w 2 .
[Solved by means of the Porisms that, if a be the
given number, the numbers m 2 — a, (m+1) 2 — a satisfy
the conditions of V. 3, and the numbers m 2 + a,
(m + l) 2 + a the conditions of V. 4 (see p. 479 above). The
third number is taken to be 2 {m 2 + a + (m + l) 2 + a} — 1,
and the three numbers automatically satisfy two more
conditions (see p. 480 above). It only remains to make
2 {m 2 + a + (m + 1) 2 + a] — 1 +a & square,
or 4 m 2 + 4m + 3 a + 1 = a square,
which is easily solved.
With Diophantus £ + 3 takes the place of m in V. 3
and £ takes its place in V. 4, while a is 5 in V. 3 and 6
in V. 4.]
V. 5. y 2 z 2 + x 2 = r 2 , z 2 x 2 + y 2 = s 2 , x 2 y 2 + z 2 = t 2 ,
y 2 z 2 + y 2 + z 2 — u 2 , z 2 x 2 + z 2 + x 2 — v 2 , x 2 y 2 + x 2 + y 2 = w 2
[Solved by means of the Porism numbered 2 on p. 480.
K k 2
500 DIOPHANTUS OF ALEXANDRIA
V. 6. x-2 = r 2 , y-2 = s 2 , z-2 = t 2 ,
yz — y — z = u 2 , zx — z — x = v 2 , xy — x — y = %v 2 ,
yz — x — w' 2 , zx — y — v' 2 , xy — z = w' 2 .
[Solved by means of the proposition numbered (3) on
p. 481.]
Lemma 1 to V. 7. xy + x 2 -\-y 2 = u 2 .
(u 2 (v-
V. 7. x 2 ±{x + y + z)= , 2 , y 2 ±(x + y + z) = ,
(v' 2
z 2 ±{x + y + z) =
'w 2
w' 2
[Solved by means of the subsidiary problem (Lemma 2)
of finding three rational right-angled triangles with
equal area. If m, n satisfy the condition in Lemma 1,
i. e. if mn + on 2 + n 2 = p 2 , the triangles are ' formed ' from
the pairs of numbers (p, m), (p, n), (p,m + n) respec-
tively. Diophantus assumes this, but it is easy to prove.
In his case m = 3, n £= 5, so that p = 7. Now, in
a right-angled triangle, (hypotenuse) 2 + four times area
is a square. We equate, therefore, x + y + z to four
times the common area multiplied by £ 2 , and the several
numbers x, y, z to the three hypotenuses multiplied by £,
and equate the two values. In Diophantus's case the
triangles are (40, 42, 58), (24, 70, 74) and (15, 112, 113),
and 245£ = 3360£ 2 .]
lu 2 (v 2
V. 8. yz±(x + y + z)= j , 2 > zx±(x+y + z) = j , 2 >
xy± ( x + y + z ) = |^ 2 .
[Solved by means of the same three rational right-
angled triangles found in the Lemma to V. 7, together
with the Lemma that we can solve the equations yz—a 2 ,
zx = b 2 , xy = c 2 .]
V. 9. (Cf. II. 11). x + y = 1, x + a = u 2 , y + a = v 2 .
V. 11. x + y + z= 1, x + a — u 2 , y + a=v 2 , z + a — w 2 .
[These are the problems of 7rapio-6Tr]Tos dyooyrj
INDETERMINATE ANALYSIS 501
described above (pp. 477-9). The problem is ' to divide
unity into two (or three) parts such that, if one and the
same given number be added to each part, the results are
all squares '.]
(V. 10. x + y — 1, x + a = u 2 , y + b = v 2 .
\V. 12. x+y+z— 1, x + a = 16 2 , y + b = v 2 , z + c = w 2 .
[These problems are like the preceding except that
different given numbers are added. The second of the
two problems is not worked out, but the first is worth
reproducing. We must take the particular figures used
by Diophantus, namely a = 2, b — 6. We have then to
divide 9 into two squares such that one of them lies
between 2 and 3. Take two squares lying between 2
and 3, say f J J, fff . We have then to find a square £ 2
lying between them ; if we can do this, we can make
9 — £ 2 a square, and so solve the problem.
Put 9-£ 2 = (3-m£) 2 , say, so that £ = 6m/(m 2 + 1) ;
and m has to be determined so that £ lies between
T2 anc l T§ •
rnu e 17 6m 19
Inereiore — < — ^ < — •
12 m 2 +l 12
Diophantus, as we have seen, finds a fortiori integral
limits for m by solving these inequalities, making m not
greater than ff and not less than J § (see pp. 463-5 above).
He then takes m = 3J and puts 9 — £ 2 = (3 — 3J£) 2 ,
which gives £ = f§.]
V. 13. x + y + z = a, y + z — u 2 , z + x = v 2 , x + y = w 2 .
V. 14. x + y + z + w = a, x + y + z = s 2 , y + z + w=t 2 ,
z + w + x — u 2 i w + x + y — v 2 .
[The method is the same.]
V. 21. x 2 y 2 z 2 + x 2 = u 2 , x 2 y 2 z 2 + y 2 = v 2 , x 2 y 2 z 2 + z 2 = iv 2 .
V. 22. x 2 y 2 z 2 -x 2 = < x 2 y 2 z 2 -y 2 = v 2 , x 2 y 2 z 2 -z 2 = w 2 .
V. 23. x 2 ~x 2 y 2 z 2 = u 2 , y 2 -x 2 y 2 z 2 = v 2 , z 2 -x 2 y 2 z 2 = iv 2 .
[Solved by means of right-angled triangles in rational
numbers.]
502 DIOPHANTUS OF ALEXANDRIA
(V. 24. y 2 z 2 + 1 = u 2 , z 2 x 2 +\ ~ v 2 , x 2 y 2 + 1 = w 2 .
V. 25. y 2 z 2 - 1 = u 2 , z 2 x 2 -\ = v 2 , x 2 y 2 -\ = w 2 .
IV. 26. 1 -y 2 z 2 = t^ 2 , 1 -z 2 x 2 = v 2 , 1 -x 2 y 2 = w 2 .
[These reduce to the preceding set of three problems.]
IV. 27. y 2 + z 2 + a = u 2 , z 2 + x 2 + a = ^ 2 , a 2 + ?/ 2 + a = «*
(V. 28. y 2 + z 2 — a = u 2 , z 2 + x 2 — a = v 2 , x 2 + y 2 — a = w 2 .
V. 30. mx + ny = u 2 , u 2 + a= (x + y) 2 .
[This problem is enunciated thus. £ A man buys a
certain number of measures of wine, some at 8 drachmas,
some at 5 drachmas each. He pays for them a square
number of drachmas ; and if 60 is added to this number,
the result is a square, the side of which is equal to the
whole number of measures. Find the number bought at
each price.'
Let £ = the whole number of measures ; therefore
£ 2 — 60 was the number of drachmas paid, and £ 2 — 60
= a square, say (£— m) 2 ; hence £ = (m 2 + 60)/2m.
Now -§• of the price of the five-drachma measures + J
of that of the eight-drachma measures = £ ; therefore
g 2 — 60, the total price, has to be divided into two parts
such that -J of one + § of the other = £.
We cannot have a real solution of this unless
£> 1(^-60) and <-i(£ 2 -60);
therefore 5£ < £ 2 -60 < 8£.
Diophantus concludes, as we have seen (p. 464 above),
that £ is not less than 11 and not greater than 12.
Therefore, from above, since £ = (m 2 + 60)/2m,
22m < m 2 + 60 < 24m;
and Diophantus concludes that m is not less than 19 and
not greater than 21. He therefore puts m = 20.
Therefore £ = (m 2 + 60)/2m = 11 J, g 2 = 132J, and
£ 2 — 60 = 72J.
We have now to divide 72 J into two parts such that
| of one part + J of the other = 1 lj.
INDETERMINATE ANALYSIS
503
Let the first part = 5 z ; therefore f (second part)
= \\^ — z, or second part = 92 — Sz.
Therefore 5z + 92 - Sz = 72J, and z = \\\
therefore the number of five-drachma measures is \ § and
the number of eight-drachma measures ff .]
Lemma 2 to VI. 1 2. ax 2 + b = u 2 (where a + b — c 2 ). | / see p 457
Lemma to VI. 15. ax 2 -b=u 2 (where ad 2 -b = c 2 ).} above.)
([III. 15]. xy + x + ;>/ = u 2 , x+1 = — 2 (y+l).
[III. 16]. xy — (x + y) = ti 2 , x—l =~ z (y—\).
[IV. 32]. flj+l =^(aj_l).
[V. 21]. x 2 + 1 = u 2 , y 2 + 1 = v 2 , s 2 + l = w 2 .
IV.
(IV.
(IV.
JV.
IV.
IV.
IV.
IV.
(viii) Indeterminate analysis of the third degree.
3. x 2 y — u, xy — u 3 .
6. x 3 + y 2 = u 3 , z 2 + y 2 = v 2 .
7. x 3 -\-y 2 = u 2 , z 2 + y 2 = v 3 .
8. x + y 3 = u 3 , x + y = u.
9. x + y 3 = u, x + y = u 3 .
10. x 3 -\-y 3 = x + y.
11. x 6 — y 6 — x — y.
12. 2^ + 2/ = y 3 + x.,
the same problem.
(really reducible
to the second
degree.)
x 3 .
[We may give as examples the solutions of IV. 7
IV. 8, IV. 11.
IV. 7. Since z 2 + y 2 = a cube, suppose z 2 + y 2
To make x 3 + y 2 a square, put # 3 = a 2 + b 2 , y 2 = 2 a&,
which also satisfies x 3 — y 2 = z 2 . We have then to make
2ab & square. Let a = g, b = 2g; therefore a 2 + b 2 = 5 £ 2 ,
2a6 = 4| 2 , 2/ = 2£, # : = £, and we have only to make
5£ 2 a cube. £ = 5, and ^ 3 = 125, / = 100, s 2 = 25.
504 DIOPHANTUS OF ALEXANDRIA
IV. 8. Suppose % = £, 2/ 3 = m 3 £ 3 ; therefore u=(m+ 1)£
must be the side of the cube m 3 £ 3 + £, and
m 3 £ 2 +l = (m 3 +3m 2 + 3m+l)f.
To solve this, we must have 3 m 2 + 3 m 4- 1 (the difference
between consecutive cubes) a square. Put
3m 2 + 3m+l = (l—nm) 2 , and m = (3 -f 2n)/(n 2 — 3).
IV. 11. Assume x = (m+l)£, 2/ = ?^£> and we have
to make (3m 3 + 3m 2 + 1)£ 2 equal to 1, i.e. we have
only to make 3m 2 + 3m + 1 a square.]
IV. 18. x 3 + y = it 3 , 2/ 2 + # = v 2 .
IV. 24. a? + 2/ = a, fl?2/ = u 3 — u.
[y = a — x; therefore ax — x 2 has to be made a cube
minus its side, say (mx— l) 3 — (mx— 1).
Therefore ax — x 2 = m 3 & 3 — 3 m 2 a.* 2 + 2 mx.
To reduce this to a simple equation, we have only to
put m = |a.]
IV. 25. ^ + 2/-h0 = a, ^2/^ = { ( x —y) + ( ai — ^) + (2/ ~ ) } 3 *
(.a- > y > z)
[The cube = 8(x — s) 3 . Let x = (m+l)£,z = m£,so
that ?/ = 8£/(m 2 + m), and we have only to contrive that
8/(m 2 + m) lies between m and m + 1. Dioph. takes the
first limit 8 > m 3 + m 2 , and puts
8 = (m-f -|) 3 or m 3 + w 2 + |m + ^ r ,
whence m = § ; therefore & = §|, ?/ = §-£, £ = ■§£. O r >
multiplying by 15, we have x = 40 £, s/ = 27 £, = 25 £.
The first equation then gives £.]
rIV. 26. xy + x = u 3 , xy + y = v ?> .
llV. 27. xy — x = u z , xy — y = v 3 .
IV. 28. xy + (x + y) = u ?J , xy—(x + y) = v\
[x + y = % (u 3 — v 3 ), ^2/ = i ('M' 3 + ^ 3 ) j therefore
(x — y) 2 = \ (u 3 — 1> 3 ) 2 — 2 (u 3 + v 5 ),
which latter expression has to be made a square.
INDETERMINATE ANALYSIS 505
Diophantus assumes u = £ + 1 , v = £ — 1 , whence
i(6f+2) 2 -2(2£* + 6£)
must be a square, or
9£ 4 -4£ 3 + 6£ 2 -12£+l:=a square = (3 £ 2 -6£+l) 2 , say;
therefore 32 £ 3 = 36 £ 2 , and £ = §. Thus u, v are found,
and then x, y.
The second (alternative) solution uses the formula that
§(i*-i) + (i* T £) + i = a cube. Put x = $, y= f -|,
and one condition is satisfied. We then only have to
make £(f 2 -£) -£- (£ 2 -£) or £ 3 -2£ 2 a cube (less than
a>i.e.£ 3 -2£ 2 =(i^say.]
IV. 38. (x + y + z)x = ^u(u + 1), (x + y + z)y — v 2 ,
(x + y + z)z = w 3 , [x + y + z = t 2 ~\.
[Suppose # + 2/ + s = £ 2 ; then
_ 16 (u + 1 ) v 2 u> 3
<B "' - a| i_ ' * = ?•*= j ;
therefore £ 4 = %u(u + 1) + v 2 + w :} .
Diophantus puts 8 for iv 3 , but we may take any cube, as
m 3 ; and he assumes v 2 = (£ 2 — l) 2 , for which we might
substitute (£ 2 — 7b 2 ) 2 . We then have the triangular
number %u(u+l) = 2n 2 £ 2 — n 4: — m 3 . Since 8 times a
triangular number _^us 1 gives a square,
16w, 2 £ 2 — 8?i 4 — 8m 3 + 1 = a square = (4?i£ — /c) 2 , say,
and the problem is solved.]
V. 15. (x + y + z) 3 + x = u 3 , (x + y + z) ?J + y = v\
(x + y + z) 3 + z = w z .
[Let x + y + z = £, u 3 = m 3 £ 3 , v 3 = ?i 3 £ 3 , ^{; 3 = #> 3 | 3 ;
therefore £ = { (m 3 — 1) + (n ?J — 1) + (p 3 — 1) } £ 3 ;
and we have to find three cubes m 3 , ri*, p 3 such that
m 3 + ?i 3 +p 3 — 3 = a square. Diophantus assumes as
the sides of the cubes (k+1), (2—k), 2; this gives
506
DIOPHANTUS OF ALEXANDRIA
9 A; 2 — 9&+14 = a square = (3 k — I) 2 , say; and k is found.
Retracing our steps, we find £ and therefore x, y, z.]
V. 16. (x + y + z) 3 — x = u 3 , (x + y + z) 3 — y = v 3 ,
(x + y + z) 3 — z — it' 3 .
V. 17. a; — (a? -f y + #) 3 = u 3 , y — (x + y + zf — v 3 ,
z-(x + y + z) ?J = w 3 .
V. 18. .^ + 2/ + ^ = ^ 2 , (o? + ?/ + 0) 3 + .t = u 2 , (x + y + z) 3 + y=v 2 ,
(x + y + zf' + z = iv 2 .
[Put x + y + z = £*, x = (p 2 -l)i G , y = (q 2 -l)£«,
z = (r 2 -l)g 6 , whence £ 2 = ( 2 j 2 -1 +q 2 -l +r 2 -l)£ G , so
that p 2 — 1 + q 2 — 1 + r 2 — 1 must be made a fourth
power. Diophantus assumes r p 2 = (m 2 — 1 ) 2 , q 2 = (m + 1 ) 2 ,
r 2 = (m — 1 ) 2 , since m 4 — 2 m 2 + m 2 + 2 m + m 2 — 2 m = m 4 .]
V. 19. «? + 2/ 4- ^ = £ 2 , (x + y + z) 3 — x — u 2 ,
(x + y + z) 3 — y = v 2 , (x + y + z) 3 — z = w 2 .
V. 19a. x + y + z = t 2 , x — (x + y + z) 3 = u 2 ,
y — (x + y + z) 3 — v 2 , z — (x + y + z) 3 = w 2 .
V. 1 9. b ; c. x + y + z — a, (x + y + z) 3 ± x — u 2 ,
(x + y + z) 3 + y = v 2 , (x + y + z) 3 ±z = %u 2 .
V. 20. # + ?/ + £ = — * x — (x + y + z) 3 = u 2 .
y — (x + y + z) 3 = v 2 , z — (x + y + z) 3 = w 2 .
[IV. 8]. x — y— 1, x 3 — y 3 = u 2 .
[IV. 9,10]. ^ + 2/ 3 = ^(0 + 2/).
v-
[IV. 11]. ^ 3 -2/ 3 = ^(^-2/).
[V. 15]. x 3 + y 3 + z 3 -3 =u 2 .
[V. 16]. 3-(a5 3 + 2/ 3 + s 3 ) =u 2 .
[V. 17]. .^ 3 + 2/ 3 + 5 3 + 3 = i6 2 .
INDETERMINATE ANALYSIS 507
(ix) Indeterminate analysis of the fourth degree.
V. 29. x* + y* + z* = u 2 .
[' Why ', says Fermat, ' did not Diophantus seek two
fourth powers such that their sum is a square. This
problem is, in fact, impossible, as by my method I am
able to prove with all rigour.' No doubt Diophantus
knew this truth empirically. Let x 2 = £ 2 , y 2 = p 2 ,
z 2 = q 2 . Therefore £ 4 +p* + q* = a square = (£ 2 — r) 2 , say ;
therefore £ 2 = (r 2 — p 4 — g 4 )/2r, and we have to make
this expression a square.
Diophantus puts r = £> 2 + 4, g 2 = 4, so that the expres-
sion reduces to 8p 2 /(2p 2 + 8) or 4p 2 /(p 2 + 4). To make
this a square, let p 2 + 4 = (p + l) 2 , say ; therefore p = 1-|,
and p 2 =2%, q 2 = 4, r = 6^; or (multiplying by 4)
£> 2 =9, q 2 = 16, r = 25, which solves the problem.]
[V. 18]. ^ 2 + 2/ 2 + 2 2 -3 =u 4 .
(See above under V. 18.)
(x) Problems of constructing right-angled triangles with
sides in rational numbers and satisfying various
other conditions.
[I shall in all cases call the hypotenuse z, and the
other two sides x, y, so that the condition x 2 + y 2 = z 2
applies in all cases, in addition to the other conditions
specified.]
[Lemma to V. 7]. xy = x Y y Y = x 2 y 2 .
'VI. 1. z — x = u 3 , z — y = v z .
[Form a right-angled triangle from £, m, so that
z = £ 2 + m 2 , x = 2m£, y — £ 2 — m 2 ; thus z — y = 2m 2 ,
and, as this must be a cube, we put m = 2 ; therefore
2 — # = £ 2 — 4£ + 4 must be a cube, or £ — 2 = a cube,
say % 3 , and £ = ?i 3 + 2.]
VI. 2. s + & = u 3 , z + y = v 3 .
508
DIOPHANTUS OF ALEXANDRIA
VI. 3,. \xy + a = u 2 .
[Suppose the required triangle to be kg, p£, bg ; there-
fore \pbg + a = a square = 7i 2 £ 2 , say, and the ratio of a
to n 2 -ipb must be the ratio of a square to a square.
To find n, p, b so as to satisfy this condition, form
. „ 1
a right-angled triangle trom m, — >
i.e.
(
m- +
.1
m 5
2, 771
1
therefore \pb = m 2 —
??i-
Assume n 2 = (
2 ax 2
771+ )
011/
4ft 2 +1
77 I 2
(
; and(4a +
4a 2 + 1
m'
)/
therefore ^ 2 — \pb = 4 a +
r/ (4ft 2 4-1^
or 4ft 2 + — ^ — -j has to be made a square. Put
m-
4a 2 m 2 + ft (4ft 2 + 1) = (2 am + h) 2 , and we have a solution.
Diophantus has a =5, leading to 100m 2 + 505 = a square
= (10m + 5) 2 ; say, which gives m = - 2 g 4 - and n = -^o 3 --
h, p, b are thus determined in such a way that
hpb£ 2 + a = n 2 £ 2 gives a rational solution.]
VI. 4. \xy — a = u 2 .
VI. 5. a — \xy = u 2 .
VI. 6. J#2/ + # = a.
[Assume the triangle to be hg, r pi> b£, so that
ipb£ 2 +p£ = «> an( l f° r a rational solution of this equa-
tion we must have (ip) 2 + a>(ipb) a square. Diophantus
assumes p = 1, b = m, whence Ja77i + J or 2a77i+l
= a square.
But, since the triangle is rational, m 1 + 1 = a square.
That is, we have a double equation. Difference
= m 2 — 2 am — m (m — 2a). Put
2am+ 1 = {i(m — m — 2a)} 2 = a 2 , andjm = (a 2 — l)/2a.
The sides of the auxiliary triangle are thus determined
in such a way that the original equation in £ is solved
rationally.]
VI. 7. \xy — x — a.
or
INDETERMINATE ANALYSIS 509
VI. 8. \xy +(x + y) = a.
VI. 9. ixy-(x + y) =a.
[With the same assumptions we have in these cases
to make {%{ r p + b)} 2 + a{^pb) a square. Diophantus
assumes as before 1 , m f or the values of p, b, and obtains
the double equation
J (m + 1 ) 2 + Jam = square]
m 2 + 1 = square)
m 2 + (2 a + 2) m + 1 = square]
m 2 + 1 = square)
solving in the usual way.]
VI. 10. \xy-\-x-\-z = a.
yi. 11. %xy-(x + z) = a.
[In these cases the auxiliary right-angled triangle has
to be found such that
{ i Q 1 + V) } 2 + a (i pty — a square.
Diophantus assumes it formed from 1 , m + 1 ; thus
| (A +p) 2 = J [m 2 + 2m + 2 + m 2 + 2m} 2 = (m 2 + 2m + l) 2 ,
and a (^ £>6) = a (m + 1 ) (m 2 + 2 ??i) .
Therefore
m 4 + (a + 4)m 3 + (3a + 6)m 2 + (2a + 4)m + 1
= a square
= { 1 + (a + 2) m — m 2 } 2 , say ;
and m is found.] ,
Lemma 1 to VI. 12. x = u 2 , x — y — v 2 , \xy + y = w 2 .
fVI. 12. i xy + x = u 2 , ±xy + y = v 2 .
iVI. 13. \xy — x = u 2 , ^xy — y — v 2 .
[These problems and the two following are interesting,
but their solutions run to some length ; therefore only
one case can here be given. We will take VI. 1 2 with
its Lemma 1.
510 DIOPHANTUS OF ALEXANDRIA
Lemma 1 . If a rational right-angled triangle be formed
from m, n, the perpendicular sides are 2mn, m 2 — n 2 .
We will suppose the greater of the two to be 2mn.
The first two relations are satisfied by making m = 2 n.
Form, therefore, a triangle from £, 2£. The third con-
dition then gives 6 £ 4 + 3 £ 2 = a square or 6 £ 2 + 3 = a
square. One solution is £ = 1 (and there are an infinite
number of others to be found by means of it). If £ = 1,
the triangle is formed from 1, 2.
VI. 12. Suppose the triangle to be (kg, b£,p£). Then
(ipb)i 2 +pg = a, square = (k £) 2 , say, and £=p>/(k 2 —±pb).
This value must be such as to make (i2 j1j )£ 2 + b£ a square
also. By substitution of the value of £ we get
{bpk 2 + ±p 2 b(p-b)} /(P-ip6) 2 ;
so that bpk 2 + ^p 2 b(p—b) must be a square; or, if p,
the greater perpendicular, is made a square number,
bk 2 + ^pb(2J — b) has to be made a square. This by
Lemma 2 (see p. 467 above) can be made a square if
b + ipb{p — b) is a square. How to solve these problems,
says Diophantus, is shoivn in the Lemmas. It is not
clear how they were applied, but, in fact, his solution
is such as to make p, p — b, and b + ^j J b all squares,
namely b = 3, p = 4, h = 5.
Accordingly, putting for the original triangle 3£, 4£, 5£,
we have
6 £ 2 + 4 £ = a square )
6 £ 2 -f 3 £ = a square )
Assuming 6£ 2 + 4£ = m 2 £ 2 , we have £ = 4/(m 2 — 6), and
the second condition gives
96 12
m*-12m 2 + 36 + m^^6 ~ a SqUare '
or 1 2 m 2 + 2 4 = a square.
This can be solved, since m = 1 satisfies it (Lemma 2).
A solution is m 2 = 25, whence £ = T 4 § .]
VI. 14. \xy — z — u 2 , \xy — x — v 2 .
VI. 15. \xy-\-z = it 2 , -§#?/ + £' = v 2 .
INDETERMINATE ANALYSIS 511
[The auxiliary right-angled triangle in this case must
be such that
m 2 hp — \pb . p (h —p) is a square.
If, says Diophantus (VI. 14), we form a triangle from
the numbers X lt X 2 and suppose that p = 2X X X 2 , and if
we then divide out by (X x — X 2 ) 2 , which is equal to h ~p,
we must find a square k 2 [ = f m 2 /(X 1 — X 2 ) 2 ] such that
k 2 hp — ^pb .p is a square.
The problem, says Diophantus, can be solved if X 1} X<
m
2
2
are ' similar plane numbers ' (numbers such as ab, — ab).
n 2 }
This is stated without proof, but it can easily be verified
that, if Jc 2 = X 1 X 2 , the expression is a square. Dioph.
takes 4, 1 as the numbers, so that h 2 = 4. The equation
for m becomes
8 . 17m 2 -4 . 15 . 8 . 9 = a square,
or 136 m 2 — 4 3 2 = a square.
The solution m 2 =36 (derived from the fact that
Jc 2 = m 2 /(X 1 -X 2 f, or 4 = m 2 /3 2 )
satisfies the condition that
m 2 hp — ±pb .p(h—p) is a square.]
VI. 16. i + v = x, i/rj = y/z.
[To find a rational right-angled triangle such that the
number representing the (portion intercepted within
the triangle of the) bisector of an acute angle is rational.
3fa-£) D 71 B
Let the bisector be 5 £, the segment BD of the base 3 £,
so that the perpendicular is 4£.
Let GB = 3 n. Then AC : AB = CD : DB,
512 DIOPHANTUS OF ALEXANDRIA
so that AG = 4 (n - £). Therefore (Eucl. I. 47)
16(n 2 — 2n£ + g 2 ) = 16£ 2 + 9^ 2 ,
so that £ = 7 ri 1 / 32 n = ^w. [Dioph. has n = 1.]
r VI. 17. \xy-\-z = ti 2 , cc + 2/ + ^ = u 3 .
[Let £ be the area Jar?/, and let z — k 2 — £. Since
xy = 2£, suppose x = 2, y = £. Therefore 2 + & 2 must
be a cube. As we have seen (p. 475), Diophantus
takes (m — l) 3 for the cube and (m+1) 2 for k 2 , giving
m 3 — 3 m 2 + 3 m — 1 = m 2 + 2 m + 3, whence rn = 4. There-
fore A; = 5, and we assume \xy = £ 3 s = 25 — £, with
a? = 2, y = i as before. Then we have to make
(25-£) 2 = 4 + £ 2 , and £ = «?-.]
VI. 1 8. -!#?/ + = u 3 , x + y ±z = v 2 .
" vi. i9. 4^2/ + x — u2 > ®+y+z = v 3 '
[Here a right-angled triangle is formed from one odd
number, say 2£+l, according to the Pythagorean for-
mula m 2 + {-|(m 2 — l)} 2 = {-|(m 2 +l)} 2 , where m is an
odd number. The sides are therefore 2£+l, 2£ 2 +2|,
2 £ 2 -f 2 £ + 1 . Since the perimeter == a cube,
4£ 2 + 6£ + 2 = (4£ + 2) (£+1) = a cube.
Or, if we divide the sides by £+1, 4 £ + 2 has to be.
made a cube.
a • 1 2£ 3 + 3£ 2 + £ 2£+l
Again \xy + x = /<t i * v2 + ■ * , ., = a square,
uf+ir
i+i
which reduces to 2| + 1 = a square.
But 4 £ + 2 is a cube. We therefore put 8 for the cube,
and £=li.]
VI. 20. \xy + x — u 3 , x -I- 2/ + z — v 2 .
VI. 21. x + y + z — %i A , %xy + (x + y + z) = v z .
[Form a right-angled triangle from £, 1, i.e. (2£, £ 2 — 1,
g * + 1 ). Then 2 £ 2 + 2 £ must be a square, and £ 3 + 2 £ 2 + £
INDETERMINATE ANALYSIS 513
a cube. Put 2£ 2 + 2£ = m 2 | 2 , so that £ = 2/(m 2 -2),
and we have to make
8 8 2 2 m, 4
-, + / — o — t^to + —o — ~' or ; — 5 — ^ , a cube.
(m 2 -2) 3 (m 2 -2) 2 m 2 -2 (m 2 —2) 3
Make 2 m . a cube = n^, so that 2 m 4 = m 3 ^ 3 , and
g
m = ^n 3 : therefore £= — > and £ must be made
greater than 1 , in order that £ 2 — 1 may be positive.
Therefore 8 < n G < 16;
this is satisfied by n- G = - 7 ¥ \ 9 - or w 5 = - 2 g 7 -, and m = f|.]
VI. 22. x + y + z = u 3 , %xy + (x + y + z) = v 1 .
[(1) First seek a rational right-angled triangle such
that its perimeter and its area are given numbers,
say p, m.
Let the perpendiculars be -, 2 m £; therefore the hypo-
i
tenuse = p — - — 2m£, and (Eucl. I. 47)
2 + 4m 2 £ 2 + (p 2 + 4 m) f — 4mpg — — + 4m 2 £ 2 ,
2 /j
or p 2 + 4 m = 4 mp£ + -~- >
that is, (jo 2 + 4m)£ = ^mpg* + 2^>.
(2) In order that this may have a rational solution,
{ i {'P 2 + 4m) } 2 — %p l m must be a square,
i.e. 4 m 2 — 6^) 2 m -f ^_p 4 = a square,
or m 2 — § p 2 m + t 1 ©^ 4 = a square]
Also, by the second condition, m+p = a square)
To solve this, we must take for p some number which
is both a square and a cube (in order that it may be
possible, by multiplying the second equation by some
square, to make the constant term equal to the constant
1523.2 \j 1
514 DIOPHANTUS OF ALEXANDRIA
term in the first). Diophantus takes p = 64, making
the equations
m 2 — 61 44 m + 1048576 = a square]
m + 64 = a square)
Multiplying the second by 16384, and subtracting the two
expressions, we have as the difference m 2 — 22528m.
Diophantus observes that, if we take m, m— 22528 as
the factors, we obtain m = 7680, an impossible value for
the area of a right-angled triangle of perimeter <p = 64.
We therefore take as factors 11m, y^m— 2048, and,
equating the square of half the difference ( = fym + 1024)
to 16384771+1048576, we have m = -||| 4 .
(3) Returning to the original problem, we have to
substitute this value for ra in
(64-I-2m^ = l+4m^,
and we obtain
78848£ 2 — 8432£ + 225 = 0,
the solution of which is rational, namely £ = ^fg- (or T f g).
Diophantus naturally takes the first value, though the
second gives the same triangle.]
VI. 23. z 2 = u 2 + u, z 2 / x = v* + v.
VI. 24. z = u ?> + u, x = v 3 — v, y = w*.
[VI. 6, 7]. (ixf+%mxy = u 2 .
[VI. 8, 9]. H(x + y)} 2 + ±mxy = u 2 .
[VI. 10, 11]. {i(z + x)} 2 + ±mxy = u 2 .
[VI. 12.] y + (x—y).%xy = u 2 , x = v 2 . (x > y.)
[VI. 14, 15], u 2 zx — \xy .x(z—x) = v 2 . (u 2 < or > \xy.)
The treatise on Polygonal Numbers.
The subject of Polygonal Numbers on which Diophantus
also wrote is, as we have seen, an old one, going back to the
THE TREATISE ON POLYGONAL NUMBERS 515
Pythagoreans, while Philippus of Opus and Speusippus carried
on the tradition. Hypsicles (about 170 B.C.) is twice men-
tioned by Diophantus as the author of a ' definition ' of
a polygonal number which, although it does not in terms
mention any polygonal number beyond the pentagonal,
amounts to saying that the nth a-gon (1 counting as the
first) is
i. n{2 + (n-l)(a-2)}.
Theon of Smyrna, Nicomachus and Iamblichus all devote
some space to polygonal numbers. Nicomachus in particular
gives various rules for transforming triangles into squares,
squares into pentagons, &c.
1. If we put two consecutive triangles together, we get a square.
In fact
2. A pentagon is obtained from a square by adding to it
a triangle the side of which is 1 less than that of the square ;
similarly a hexagon from a pentagon by adding a triangle
the side of which is 1 less than that of the pentagon, and so on.
In fact
in { 2 + (n - 1) (a- 2) } + i{n— \)n
= in[2 + (n-l){(a+l)-2}].
3. Nicomachus sets out the first triangles, squares, pentagons,
hexagons and heptagons in a diagram thus :
Triangles
1
3
6
10
15
21
28
36
45
55,
Squares
1
4
9
16
25
36
49
64
81
100,
Pentagons
1
5
12
22
35
51
70
92
117
145,
Hexagons
1
6
15
28
45
66
91
120
153
190,
Heptagons
1
7
18
34
55
81
112
148
189
235,
and observes that :
Each polygon is equal to the polygon immediately above it
in the diagram plus the triangle with 1 less in its side, i.e. the
triangle in the preceding column.
Ll2
516 DIOPHANTUS OF ALEXANDRIA
4. The vertical columns are in arithmetical progression, the
common difference being the triangle in the preceding column.
Plutarch, a contemporary of Nicomachus, mentions another
method of transforming triangles into squares. Every tri-
angular number taken eight times and then increased by 1
gives a square.
•In fact, 8.£n(w+l) + l = (2?i+ l) 2 .
Only a fragment of Diophantus's treatise On Polygonal
Numbers survives. Its character is entirely different from
that of the Arithmetica. The method of proof is strictly
geometrical, and has the disadvantage, therefore, of being long
and involved. He begins with some preliminary propositions
of which two may be mentioned. Prop. 3 proves that, if a be
the first and I the last term in an arithmetical progression
of n terms, and if s is the sum of the terms, 2s = n(l + a).
Prop. 4 proves that, if 1, 1+6, 1 + 26, ... 1 + (n— l)b be an
A. P., and s the sum of the terms,
2s = n {2 + (n—l)b}.
The main result obtained in the fragment as we have it
is a generalization of the formula 8 . \n{n + 1) + 1 = (2 n + l) 2 .
Prop. 5 proves the fact stated in Hypsicles's definition and also
(the generalization referred to) that
8 P (a — 2) + (a — 4) 2 = a square,
where P is any polygonal number with a angles.
It is also proved that, if P be the nth. a-gonal number
(1 being the first),
8P(a-2) + (a-4) 2 = {2 + (2n- 1) (a-2)}
Diophantus deduces rules as follows.
1 . To find the number from its side.
{2 + (2 n-1) (a- 2) } 2 - (a- 4) 2
2
i •
P =
8(a-2)
2. To find the side from the number.
= 1 /y {8P( a-2) + (a-4) 2 }-2 v
" 2 V a — 2 )
THE TREATISE ON POLYGONAL NUMBERS 517
The last proposition, which breaks off in the middle, is :
Given a number, to find in how many ways it can be
polygonal.
The proposition begins in a way which suggests that
Diophantus first proved geometrically that, if
8P(a-2) + (a-4) 2 = {2 + (2ti-1) (a- 2) } 2 ,
then 2P = n {2+ (n— l)(a— 2)}.
Wertheim (in his edition of Diophantus) has suggested a
restoration of the complete proof of this proposition, and
I have shown (in my edition) how the proof can be made
shorter. Wertheim adds an investigation of the main pro-
blem, but no doubt opinions will continue to differ as to
whether Diophantus actually solved it.
XXI
COMMENTATORS AND BYZANTINES
We have come to the last stage of Greek mathematics ; it
only remains to include in a last chapter references to com-
mentators of more or less note who contributed nothing-
original but have preserved, among observations and explana-
tions obvious or trivial from a mathematical point of view,
valuable extracts from works which have perished, or
historical allusions which, in the absence of original docu-
ments, are precious in proportion to their rarity. Nor must
it ! be forgotten that in several cases we probably owe to the
commentators the fact that the masterpieces of the great
mathematicians have survived, wholly or partly, hf the
original Greek or at all. This may have been the case even
with the works of Archimedes on which Eutocius wrote com-
mentaries. It was no doubt these commentaries which
aroused in the school of Isidorus of Miletus (the colleague
of Anthemius as architect of Saint Sophia at Constantinople)
a new interest in the works of Archimedes and caused them
to be sought out in the various libraries or wherever they had
lain hid. This revived interest apparently had the effect of
evoking new versions of the famous works commented upon
in a form more convenient for the student, with the Doric
dialect of the original eliminated; this translation of the
Doric into] the more familiar dialect was systematically
carried out in those books only which Eutocius commented
on, and it is these versions which alone survive. Again,
Eutocius's commentary on Apollonius's Conies is extant for
the first four Books, and it is probably owing to their having
been commented on by Eutocius, as well as to their being
more elementary than the rest, that these four Books alone
SERENUS 519
survive in Greek. Tannery, as we have seen, conjectured
that, in like manner, the first six of the thirteen Books of
Diophantus's Arithmetica survive because Hypatia wrote
commentaries on these Books only and did not reach the
others.
The first writer who calls for notice in this chapter is one
who was rather more than a commentator in so far as he
wrote a couple of treatises to supplement the Conies of
Apollonius, I mean Serenus. Serenus came from Antinoeia
or Antinoupolis, a city in Egypt founded by Hadrian (a. d.
117-38). His date is uncertain, but he most probably be-
longed to the fourth century A.D., and came between Pappus
and Theon of Alexandria. He tells us himself that he wrote
a commentary on the Conies of Apollonius. 1 This has
perished and, apart from a certain proposition ' of Serenus
the philosopher, from the Lemmas ' preserved in certain manu-
scripts of Theon of Smyrna (to the effect that, if a number of
rectilineal angles be subtended at a point on a diameter of a
circle which is not the centre, by equal arcs of that circle, the
angle nearer to the centre is always less than the angle more
remote), we have only the two small treatises by him entitled
On the Section of a Cylinder and On the Section of a Cone.
These works came to be connected, from the seventh century
onwards, with the Conies of Apollonius, on account of the
affinity of the subjects, and this no doubt accounts for their
survival. They were translated into Latin by Commandinus
in 1566 ; the first Greek text was brought out by Halley along
with his Apollonius (Oxford 1710), and we now have the
definitive text edited by Heiberg (Teubner 1896).
(a) On the Section of a Cylinder.
The occasion and the object of the tract On the Section of
a Cylinder are stated in the preface. Serenus observes that
many persons who were students of geometry were under the
erroneous impression that the oblique section of a cylinder
was different from the oblique section of a cone known as an
ellipse, whereas it is of course the same curve. Hence he
thinks it necessary to establish, by a regular geometrical
1 Serenus, Opuscula, ed. Heiberg, p. 52. 25-6.
520 COMMENTATORS AND BYZANTINES
proof, that the said oblique sections cutting all the generators
are equally ellipses whether they are sections of a cylinder or
of a cone. He begins with ' a more general definition ' of a
cylinder to include any oblique circular cylinder. ' If in two
equal and parallel circles which remain fixed the diameters,
while remaining parallel to one another throughout, are moved
round in the planes of the circles about the centres, which
remain fixed, and if they carry round with them the straight line
joining their extremities on the same side until they bring it
back again to the same place, let the surface described by the
straight line so carried round be called a cylindrical surface!
The cylinder is the figure contained by the parallel circles and
the cylindrical surface intercepted by them ; the parallel
circles are the bases, the axis is the straight line drawn
through their centres; the generating straight line in any
position is a side. Thirty-three propositions follow. Of these
Prop. 6 proves the existence in an oblique cylinder of the
parallel circular sections subcontrary to the series of which
the bases are two, Prop. 9 that the section by any plane not
parallel to that of the bases or of one of the subcontrary
sections but cutting all the generators is not a circle ; the
next propositions lead up to the main results, namely those in
Props. 14 and 16, where the said section is proved to have the
property of the ellipse which we write in the form
QV 2 :PV.P'V = CD 2 :CP 2 ,
and in Prop. 17, where the property is put in the Apollonian
form involving the latus rectum, QV 2 = PV . VR (see figure
on p. 137 above), which is expressed by saying that the square
on the semi-ordinate is equal to the rectangle applied to the
latus rectum PL, having the abscissa PV as breadth and falling
short by a rectangle similar to the rectangle contained by the
diameter PP f and the latus rectum PL (which is determined
by the condition PL . PP'= DD' 2 and is drawn at right angles
to PV). Prop. 18 proves the corresponding property with
reference to the conjugate diameter DD' and the correspond-
ing latus rectum t and Prop. 19 gives the main property in the
form QV 2 :PV.P'V = Q'V' 2 :PV. P'V. Then comes the
proposition that ' it is possible to exhibit a cone and a cylinder
which are alike cut in one and the same ellipse ' (Prop. 20).
SERENUS 521
Serenus then solves such problems as these : Given a cone
(or cylinder) and an ellipse on it, to find the cylinder (cone)
which is cut in the same ellipse as the cone (cylinder)
(Props. 21, 22); given a cone (cylinder), to find a cylinder
(cone) and to cut both by one and the same plane so that the
sections thus made shall be similar ellipses (Props. 23, 24).
Props. 27, 28 deal with similar elliptic sections of a scalene
cylinder and cone ; there are two pairs of infinite sets of these
similar to any one given section, the first pair being those
which are parallel and subcontrary respectively to the given
section, the other pair subcontrary to one another but not to
either of the other sets and having the conjugate diameter
occupying the corresponding place to the transverse in the
other sets, and vice versa.
In the propositions (29-33) from this point to the end of
the book Serenus deals with what is really an optical pro-
blem. It is introduced by a remark about a certain geometer,
Peithon by name, who wrote a tract on the subject of
parallels. Peithon, not being satisfied with Euclid's treat-
ment of parallels, thought to define parallels by means of an
illustration, observing that parallels are such lines as are
shown on a wall or a roof by the shadow of a pillar with
a light behind it. This definition, it appears, was generally
ridiculed ; and Serenus seeks to rehabilitate Peithon, who
was his friend, by showing that his statement is after all
mathematically sound. He therefore proves, with regard to
the cylinder, that, if any number of rays from a point outside
the cylinder are drawn touching it on both sides, all the rays
pass through the sides of a parallelogram (a section of the
cylinder parallel to the axis) — Prop. 29 — and if they are
produced farther to meet any other plane parallel to that
of the parallelogram the points in which they meet the plane
will lie on two parallel lines (Prop. 30) ; he adds that the lines
will not seem parallel (vide Euclid's Optics, Prop. 6). The
problem about the rays touching the surface of a cylinder
suggests the similar one about any number of rays from an
external point touching the surface of a cone ; these meet the
surface in points on a triangular section of the cone (Prop. 32)
and, if produced to meet a plane parallel to that of the
triangle, meet that plane in points forming a similar triangle
522 COMMENTATORS AND BYZANTINES
(Prop. 33). Prop. 31 preceding these propositions is a par-
ticular case of the constancy of the anharmonic ratio of a
pencil of four rays. If two sides AB, AC of a triangle meet
a transversal through D, an external point, in E, F and another
ray AG between AB and AG cuts DEF in a point G such
that ED : DF = EG : GF, then any other transversal through
D meeting AB, AG, AG in K, L, M is also divided harmoni-
cally, i.e. KB : DM = KL : LM. To prove the succeeding pro-
positions, 32 and 33, Serenus uses this proposition and a
reciprocal of it combined with the harmonic property of the
pole and polar with reference to an ellipse.
(f3) On the Section of a Gone.
The treatise On the Section of a Cone is even less important,
although Serenus claims originality for it. It deals mainly
with the areas of triangular sections of right or scalene cones
made by planes passing through the vertex and either through
the axis or not through the axis, showing when the area of
a certain triangle of a particular class is a maximum, under
what conditions two triangles of a class may be equal in area,
and so on, and solving in some easy cases the problem of
finding triangular sections of given area. This sort of investi-
gation occupies Props. 1-57 of the work, these propositions
including various lemmas required for the proofs of the
substantive theorems. Props. 58-69 constitute a separate
section of the book dealing with the volumes of right cones
in relation to their heights, their bases and the areas of the
triangular sections through the axis.
The essence of the first portion of the book up to Prop. 57
is best shown by means of modern notation. We will call h
the height of a right cone, r the radius of the base ; in the
case of an oblique cone, let p be the perpendicular from the
vertex to the plane of the base, d the distance of the foot of
this perpendicular from the centre of the base, r the radius
of the base.
Consider first the right cone, and let 2 x be the base of any
triangular section through the vertex, while of course 2r is
the base of the triangular section through the axis. Then, if
A be the area of the triangular section with base 2x,
A = x V (r 2 — x 2 + h 2 ).
SERENUS 523
Observing that the sum of x 2 and r 2 — x 2 + h 2 is constant, we
see that A 2 , and therefore A, is a maximum when
x 2 = r 2 - x 2 + Ir, or x 2 = J (r 2 + li 2 ) ;
and, since x is not greater than r, it follows that, for a real
value of x (other than v), h is less than r, or the cone is obtuse-
angled. When h is not less than r, the maximum triangle is
the triangle through the axis and vice versa (Props. 5,8);
when k = r, the maximum triangle is also right-angled
(Prop. 13).
If the triangle with base 2 c is equal to the triangle through
the axis, h 2 r 2 = c 2 (r 2 — c 2 + h 2 ) } or (r 2 — c 2 ) (c 2 — h 2 ) = 0, and,
since c<r, h = c, so that h<r (Prop. 10). If x lies between r
and c in this case, (r 2 — x 2 ) (x 2 — h 2 ) > or x 2 (r 2 — x 2 + h 2 ) >h 2 r 2 ,
and the triangle with base 2x is greater than either of the
equal triangles with bases 2r, 2c, or 2 h (Prop. 11).
In the case of the scalene cone Serenus compares individual
triangular sections belonging to one of three classes with other
sections of the same class as regards their area. The classes
are :
(1) axial triangles, including all sections through the axis;
(2) isosceles sections, i.e. the sections the bases of which are
perpendicular to the projection of the axis of the cone on the
plane of the base ;
(3) a set of triangular sections the bases of which are (a) the
diameter of the circular base which passes through the foot of
the perpendicular from the vertex to the plane of the base, and
(6) the chords of the circular base parallel to that diameter.
After two preliminary propositions (15, 16) and some
lemmas, Serenus compares the areas of the first class of
triangles through the axis. If, as we said, p is the perpen-
dicular from the vertex to the plane of the base, d the distance
of the foot of this perpendicular from the centre of the base,
and 6 the angle which the base of any axial triangle with area
A makes with the base of the axial triangle passing through
p the perpendicular,
A =?V(£> 2 + d 2 sin 2 <9).
This area is a minimum when = 0, and increases with
524 COMMENTATORS AND BYZANTINES
until 6 = \tt when it is a maximum, the triangle being then
isosceles (Prop. 24).
In Prop. 29 Serenus takes up the third class of sections with
bases parallel to d. If the base of such a section is 2x,
A=xV(r 2 -x 2 +p 2 )
and, as in the case of the right cone, we must have for a real
maximum value
x 2 = \ (r 2 +2 j2 )> while x<r t
so that, for a real value of x other than r, p must be less than
r 3 and, if p is not less than r, the maximum triangle is that
which is perpendicular to the base of the cone and has 2 r for
its base (Prop. 29). If p<r, the triangle in question is not
the maximum of the set of triangles (Prop. 30).
Coming now to the isosceles sections (2), we may suppose
2 6 to be the angle subtended at the centre of the base by the
base of the section in the direction away from the projection
of the vertex. Then
A = r sin 6 V { p l + (d + r cos 6) 2 } .
If A be the area of the isosceles triangle through the axis,
we have
A 2 -A 2 = r 2 (p 2 + d 2 ) - t 2 sin 2 6 (p 2 + d? + r 2 cos 2 + 2dr cos 6)
— r 2 (^2 _|_ ^2j cos 2 Q _ r 4 gi n 2 cog 2 Q _ 2 (fo^Qg Q s ^ n 2 Q
If A = A 0J we must have for triangles on the side of the
centre of the base of the cone towards the vertex of the cone
(since cos is negative for such triangles)
p 2 + d 2 < r 2 sin 2 6, and a fortiori p 2 + d 2 < r 2 (Prop. 35).
If p> 2 + d 2 zir 2 , A is always greater than A, so that A is the
maximum isosceles triangle of the set (Props. 31, 32).
If A is the area of any one of the isosceles triangles with
bases on the side of the centre of the base of the cone away
from the projection of the vertex, cos is positive and A is
proved to be neither the minimum nor the maximum triangle
of this set of triangles (Props. 36, 40-4).
In Prop. 45 Serenus returns to the set of triangular sections
through the axis, proving that the feet, of the perpendiculars
from the vertex of the cone on their bases all lie on a circle
the diameter of which is the straight line joining the centre of
SERENUS 525
the base of the cone to the projection of the vertex on its
plane ; the areas of the axial triangles are therefore propor-
tional to the generators of the cone with the said circle as
base and the same vertex as the original cone. Prop. 50 is to
the effect that, if the axis of the cone is equal to the radius of
the base, the least axial triangle is a mean proportional
between the greatest axial triangle and the isosceles triangular
section perpendicular to the base ; that is, with the above nota-
tion, if r = V(p 2 + d 2 ), then r \/{p 2 + d 2 ) :rp = rp:p </(r 2 — d 2 ),
which is indeed obvious.
Prop, 57 is interesting because of the lemmas leading to it.
It proves that the greater axial triangle in a scalene cone has
the greater perimeter, and conversely. This is proved by
means of the lemma (Prop. 54), applied to the variable sides
of axial triangles, that if a 2 + d 2 = b 2 + c 2 and a>b^.od,
then a + d < b + c (a,d are the sides other than the base of one
axial triangle, and b, c those of the other axial triangle com-
pared with it; and if ABC, ADEbe two axial triangles and
the centre of the base, BA 2 + AC 2 =DA 2 + AE 2 because each
of these sums is equal to 2 A 2 + 2 BO 2 , Prop. 1 7). This proposi-
tion again depends on the lemma (Props. 52, 53) that, if
straight lines be ' inflected ' from the ends of the base of
a segment of a circle to the curve (i. e. if we join the ends
of the base to any point on the curve) the line (i. e. the sum of
the chords) is greatest when the point taken is the middle
point of the arc, and diminishes as the point is taken farther
and farther from that point.
Let B be the middle point of the
arc of the segment ABC, D, E any
other points on the curve towards
G\ I say that
AB + BC>AD + DG>AE+EC.
With B as centre and BA as radius
describe a circle, and produce AB,
AD, AE to meet this circle in F, G,
H. Join FG, GC, HG
Since AB = BG = BF, we have AF = AB + BG Also the
angles BFC, BGF are equal, and each of them is half of
the angle ABG.
526 COMMENTATORS AND BYZANTINES
Again lAGC = I AFC = \LABC = \LADC;
therefore the angles DGC, DCG are equal and DG — DC;
therefore AG = AD + DC.
Similarly EH = EC and All = AE+ EC.
But, by Eucl. III. 7 or 15, AF>AG >AH, and so on ;
therefore AB + BC> AD + DC>AE+ EC, and so on.
In the particular case where the segment ABC is a semi-
circle AB 2 + BC 2 = AC 2 = AD 2 + DC 2 , &c, and the result of
Prop. 57 follows.
Props. 58-69 are propositions of this sort: In equal right
cones the triangular sections through the axis are reciprocally
proportional to their bases and conversely (Props. 58, 59) ;
right cones of equal height have to one another the ratio
duplicate of that of their axial triangles (Prop. 62); right
cones which are reciprocally proportional to their bases have
axial triangles which are to one another reciprocally in the
triplicate ratio of their bases and conversely (Props. 66, 67);
and so on.
Theon of Alexandria lived towards the end of the fourth
century A.D. Suidas places him in the reign of Theodosius I
(379-95); he tells us himself that he observed a solar eclipse
at Alexandria in the year 365, and his notes on the chrono-
logical tables of Ptolemy extend down to 372.
Commentary on the Syntaxis.
We have already seen him as the author of a commentary
on Ptolemy's Syntaxis in eleven Books. This commentary is
not calculated to give us a very high opinion of Theon's
mathematical calibre, but it is valuable for several historical
notices that it gives, and we are indebted to it for a useful
account of the Greek method of operating with sexagesimal
fractions, which is illustrated by examples of multiplication,
division, and the extraction of the square root of a non-square
number by way of approximation. These illustrations of
numerical calculation have already been given above (vol. i,
THEON OF ALEXANDRIA 527
pp. 58-63). Of the historical notices we may mention the
following. (1) Theon mentions the treatise of Menelaus On
Chords in a Circle, i. e. Menelaus's Table of Chords, which came
between the similar Tables of Hipparchus and Ptolemy. (2) A
quotation from Diophantus furnishes incidentally a lower limit
for the date of the Arithmetica. (3) It is in the commentary
on Ptolemy that Theon tells us that the second part of Euclid
VI. 33 relating to sectors in equal circles was inserted by him-
self in his edition of the Elements, a notice which is of capital
importance in that it enables the Theonine manuscripts of
Euclid to be distinguished from the ante-Theonine, and is
therefore the key to the question how far the genuine text
of Euclid was altered in Theon's edition. (4) As we have
seen (pp. 207 sq.), Theon, a propos of an allusion of Ptolemy
to the theory of isoperimetric figures, has preserved for us
several propositions from the treatise by Zenodorus on that
subject.
Theon's edition of Euclid's Elements.
We are able to judge of the character of Theon's edition of
Euclid by a comparison between the Theonine manuscripts
and the famous Vatican MS. 190, which contains an earlier
edition than Theon's, together with certain fragments of
ancient papyri. It appears that, while Theon took some
trouble to follow older manuscripts, it was not so much his
object to get the most authoritative text as to make what he
considered improvements of one sort or other, (l) He made
alterations where he found, or thought he found, mistakes in
the original; while he tried to remove some real blots, he
altered other passages too hastily when a little more considera-
tion would have shown that Euclid's words are right or could
be excused, and offer no difficulty to an intelligent reader.
(2) He made emendations intended to improve the form or
diction of Euclid ; in general they were prompted by a desire
to eliminate anything which was out of the common in expres-
sion or in form, in order to reduce the language to one and the
same standard or norm. (3) He bestowed, however, most
attention upon additions designed to supplement or explain
the original ; (a) he interpolated whole propositions where he
thought them necessary or useful, e.g. the addition to VI. 33
528 COMMENTATORS AND BYZANTINES
already referred to, a second case to VI. 27, a porism or corollary
to II. 4, a second porism to III. 16, the proposition VII. 22,
a lemma after X. 12, besides alternative proofs here and there ;
(b) he added words for the purpose of making smoother and
clearer, or more precise, things which Euclid had expressed
with unusual brevity, harshness, or carelessness ; (c) he sup-
plied intermediate steps where Euclid's argument seemed too
difficult to follow. In short, while making only inconsider-
able additions to the content of the Elements, he endeavoured
to remove difficulties that might be felt b}^ learners in study-
ing the book, as a modern editor might do in editing a classical
text-book for use in schools ; and there is no doubt that his
edition was approved by his pupils at Alexandria for whom it
was written, as well as by later Greeks, who used it almost
exclusively, with the result that the more ancient text is only
preserved complete in one manuscript.
Edition of the Optics of Euclid.
In addition to the Elements, Theon edited the Optics of
Euclid ; Theon's recension as well as the genuine work is
included by Heiberg in his edition. It is possible that the
Catoptrica included by Heiberg in the same volume is also by
Theon.
Next to Theon should be mentioned his daughter Hypatia,
who is mentioned by Theon himself as having assisted in the
revision of the commentary on Ptolemy. This learned lady
is said to have been mistress of the whole of pagan science,
especially of philosophy and medicine, and by her eloquence
and authority to have attained such influence that Christianity
considered itself threatened, and she was put to death by
a fanatical mob in March 415. According to Suidas she wrote
commentaries on Diophantus, on the Astronomical Canon (of
Ptolemy) and on the Conies of Apollonius. These works
have not survived, but it has been conjectured (by Tannery)
that the remarks of Psellus (eleventh century) at the begin-
ning of his letter about Diophantus, Anatolius, and the
Egyptian method of arithmetical reckoning were taken bodily
from some manuscript of Diophantus containing an ancient
and systematic commentary which may very well have been
that of Hypatia. Possibly her commentary may have extended
HYPATIA. PORPHYRY 529
•
only to the first six Books, in which case the fact that Hypatia
wrote a commentary on them may account for the survival of
these Books while the rest of the thirteen were first forgotten
and then lost.
It will be convenient to take next the series of Neo-
Platonist commentators. It does not appear that Ammonius
Saccas (about a.d. 175-250), the founder of. Neo-Platonism, or
his pupil Plotinus (a.d. 204-69), who first expounded the
doctrines in systematic form, had any special connexion with
mathematics, but Porphyry (about 232-304), the disciple of
Plotinus and the reviser and editor of his works, appears to
have written a commentary on the Elements. This we gather
from Proclus, who quotes from Porphyry comments on Eucl.
I. 14 and 26 and alternative proofs of I. 18, 20. It is possible
that Porphyry's work may have been used later by Pappus in
writing his own commentary, and Proclus may have got his
references from Pappus, but the form of these references sug-
gests that he had direct access to the original commentary of
Porphyry.
Iamblichus (died about a.d. 330) was the author of a com-
mentary on the Introductio arithmetica of Nicomachus, and
of other works which have already been mentioned. He was
a pupil of Porphyry as well as of Anatolius, also a disciple of
Porphyry.
But the most important of the Neo-Platonists to the his-
torian of mathematics is Proclus (a.d. 410-85). Proclus
received his early training at Alexandria, where Olympio-
dorus was his instructor in the works of Aristotle, and
mathematics was taught him by one Heron (of course a
different Heron from the ' mechanicus Hero' of the Metrica,
&c). He afterwards went to Athens, where he learnt the
Neo-Platonic philosophy from Plutarch, the grandson of Nes-
torius, and from his pupil Syrianus, and became one of its
most prominent exponents. He speaks everywhere with the
highest respect of his masters, and was in turn regarded with
extravagant veneration by his contemporaries, as we learn
from Marinus, his pupil and biographer. On the death of
Syrianus he was put at the head of the Neo-Platonic school.
He was a man of untiring industry, as is shown by the
1523.2 M m
530 COMMENTATORS AND BYZANTINES
number of books which he wrote, including a large number of
commentaries, mostly on the dialogues of PJato (e.g. the
Timaeus, the Republic, the Parmenides, the Cratylus). He
was an acute dialectician and pre-eminent among his contem-
poraries in the range of his learning; he was a competent
mathematician ; he was even a poet. At the same time he
was a believer in all sorts of myths and mysteries, and
a devout worshipper of divinities both Greek and Oriental.
He was much more a philosopher than a mathematician. In
his commentary on the Timaeus, when referring to the ques-
tion whether the sun occupies a middle place among the
planets, he speaks as no real mathematician could have
spoken, rejecting the view of Hipparchus and Ptolemy because
6 Qzovpyos (sc. the Chaldean, says Zeller) thinks otherwise,
' whom it is not lawful to disbelieve '. Martin observes too,
rather neatly, that ' for Proclus the Elements of Euclid had
the good fortune not to be contradicted either by the Chaldean
Oracles or by the speculations of Pythagoreans old and new '.
Commentary on Euclid, Book I.
For us the most important work of Proclus is his commen-
tary on Euclid, Book I, because it is one of the main sources
of our information as to the history of elementary geometry.
Its great value arises mainly from the fact * that Proclus had
access to a number of historical and critical works which are
now lost except for fragments preserved by Proclus and
others.
(a) Sources of the Commentary.
The historical work the loss of which is most deeply to be
deplored is the History of Geometry by Eudemus. There
appears to be no reason to doubt that the work of Eudemus
was accessible to Proclus at first hand. For the later writers
Simplicius and Eutocius refer to it in terms such as leave no
doubt that they had it before them. Simplicius, quoting
Eudemus as the best authority on Hippocrates's quadratures
of lunes, says he will set out what Eudemus says * word for
word ', adding only a little explanation in the shape of refer-
ences to Euclid's Elements 'owing to the memorandum-like
style of Eudemus, who sets out his explanations in the abbre-
PROCLUS 531
viated form usual with ancient writers. Now in the second
book of the history of geometry he writes as follows '} In
like manner Eutocius speaks of the paralogisms handed down
in connexion with the attempts of Hippocrates and Antiphon
to square the circle, 'with which I imagine that all persons
are accurately acquainted who have examined {kTrz<JKtii\xkv ov ?)
the geometrical history of Eudemus and know the Geria
Aristotelica \ 2
The references by Proclus to Eudemus by name are not
indeed numerous ; they are five in number ; but on the other
hand he gives at least as many other historical data which can
with great probability be attributed to Eudemus.
Proclus was even more indebted to Geminus, from whom
he borrows long extracts, often mentioning him by name —
there are some eighteen such references — but often omitting
to do so. We are able to form a tolerably certain judge-
ment as to the origin of the latter class of passages on the
strength of the similarity of the subjects treated and the views
expressed to those found in the acknowledged extracts. As
we have seen, the work of Geminus mainly cited seems to
have borne the title The Doctrine or Theory of the Mathematics,
which was a very comprehensive work dealing, in a portion of
it, with the ' classification of mathematics '.
We have already discussed the question of the authorship
of the famous historical summary given by Proclus. It is
divided, as every one knows, into two distinct parts between
which comes the remark, ' Those who compiled histories
bring the development of this science up to this point. Not
much younger than these is Euclid, who ', &c. The ultimate
source at any rate of the early part of the summary must
presumably have been the great work of Eudemus above
mentioned.
It is evident that Proclus had before him the original works
of Plato, Aristotle, Archimedes and Plotinus, the ^v/ifiLKra of
Porphyry and the works of his master Syrianus, as well as a
group of works representing the Pythagorean tradition on its
mystic, as distinct from its mathematical, side, from Philo-
laus downwards, and comprising the more or less apocryphal
1 Simplicius on Arist. Phys., p. 60. 28, Diels.
2 Archimedes, ed. Heib., vol. iii, p. 228. 17-19.
M m 2
532 COMMENTATORS AND BYZANTINES
Upbs \6y09 of Pythagoras, the Oracles (\6yia) and Orphic
verses.
The following will be a convenient summary of the other
works used by Proclus, and will at the same time give an
indication of the historical value of his commentary on
Euclid, Book I :
Budemus : History of Geometry.
Geminus: The Theory of the Mathematical Sciences.
Heron : Commentary on the Elements of Euclid.
Porphyry:
Pappus:
Apollonius of Perga : A work relating to elementary
geometry.
Ptolemy : On the parallel-postulate.
Posidonius : A book controverting; Zeno of Sidon.
Carpus : Astronomy. '
Syrianus : A discussion on the a rigle.
(f3) Character of the Commentary.
We know that in the Neo-Platonic school the pupils learnt
mathematics ; and it is clear that Proclus taught this subject,
and that this was the origin of his commentary. Many
passages show him as a master speaking to scholars ; in one
place he speaks of ' my hearers \ l Further, the pupils whom
he was addressing were beginners in mathematics ; thus in one
passage he says that he omits ' for the present ' to speak of the
discoveries of those who employed the curves of Nicomedes
and Hippias for trisecting an angle, and of those who used the
Archimedean spiral for dividing an angle in a given ratio,
because these things would be ' too difficult for beginners \ 2
But there are signs that the commentary was revised and
re-edited for a larger public ; he speaks for instance in one
place of ' those who will come across his work \ 3 There are
also passages, e.g. passages about the cylindrical helix, con-
choids and cissoids, which would not have been understood by
the beginners to whom he lectured.
1 Proclus on Eucl. T, p. 210. 19. - lb., p. 272. 12.
3 lb., p. 84. 9.
PROCLUS 533
The commentary opens with two Prologues. The first is
on mathematics in general and its relation to, and use in,
philosophy, from which Proclus passes to the classification of
mathematics. Prologue II deals with geometry generally and
its subject-matter according to Plato, Aristotle and others.
After this section comes the famous summary (pp. 64-8)
ending with a eulogium of Euclid, with particular reference
to the admirable discretion shown in the selection of the pro-
positions which should constitute the Elements of geometry,
the ordering of the whole subject-matter, the exactness and
the conclusiveness of the demonstrations, and the power with
which every question is handled. Generalities follow, such as
the discussion of the nature of elements, the distinction between
theorems and problems according to different authorities, and
finally a division of Book I into three main sections, (1) the
construction and properties of triangles and their parts and
the comparison between triangles in respect of their angles
and sides, (2) the properties of parallels and parallelograms
and their construction from certain data, and (3) the bringing
of triangles and parallelograms into relation as regards area.
Coming to the Book itself, Proclus deals historically and
critically with all the definitions, postulates and axioms in
order. The notes on the postulates and axioms are preceded
by a general discussion of the principles of geometry, hypo-
theses, postulates and axioms, and their relation to one
another ; here as usual Proclus quotes the opinions of all the
important authorities. Again, when he comes to Prop. 1, he
discusses once more the difference between theorems and
problems, then sets out and explains the formal divisions of
a proposition, the enunciation (rrporacrLs), the setting-out
(eKOeais), the definition or specification (8io pianos), the con-
struction (Karaa-Kevrj), the 2 jroo f (dTToSeigts), the conclusion
(av/jL7repaor/ia), and finally a number of other technical terms,
e.g. things said to be given, in the various senses of this term,
the lemma, the case, the porism in its two senses, the objection
(evo-Tacris), the reduction of a problem, reductio ad absurdum,
analysis and synthesis.
In his comments on the separate propositions Proclus
generally proceeds in this way : first he gives explanations
regarding Euclid's proofs, secondly he gives a few different
534 COMMENTATORS AND BYZANTINES
cases, mainly for the sake of practice, and thirdly he addresses
himself to refuting objections which cavillers had taken or
might take to particular propositions or arguments. He does
not seem to have had any notion of correcting or improving
Euclid; only in one place does he propose anything of his
own to get over a difficulty which he finds in Euclid ; this is
where he tries to prove the parallel-postulate, after giving
Ptolemy's attempt to prove it and pointing out objections to
Ptolemy's proof.
The book is evidently almost entirely a compilation, though
a compilation ' in the better sense of the term '. The onus
probandi is on any one who shall assert that anything in it is
Proclus's own ; very few things can with certainty be said to
be so. Instances are (1) remarks on certain things which he
quotes from Pappus, since Pappus was the last of the com-
mentators whose works he seems to have used, (2) a defence
of Geminus against Carpus, who criticized Geminus's view of
the difference between theorems and problems, and perhaps
(3) criticisms of certain attempts by Apollonius to improve on
Euclid's proofs and constructions ; but the only substantial
example is (4) the attempted proof of the parallel-postulate,
based on an ' axiom ' to the effect that, ' if from one point two
straight lines forming an angle be produced ad infinitum, the
distance between them when so produced ad infinitum exceeds
any finite magnitude (i. e. length) ', an assumption which
purports to be the equivalent of a statement in Aristotle. 1
Philoponus says that Proclus as well as Ptolemy wrote a whole
book on the parallel-postulate. 2
It is still not quite certain whether Proclus continued his
commentaries beyond Book I. He certainly intended to do so,
for, speaking of the trisection of an angle by means of certain
curves, he says, ' we may perhaps more appropriately examine
these things on the third Book, where the writer of the
Elements bisects a given circumference ', and again, after
saying that of all parallelograms which have the same peri-
meter the square is the greatest ' and the rhomboid least of
all', he adds, ' But this we will prove in another place, for it
is more appropriate to the discussion of the hypotheses of the
1 De caelo, i. 5, 271 b 28-30.
2 Philoponus on Anal. Post. i. 10, p. 214 a 9-12, Brandis.
PROCLUS 535
second Book \ But at the time when the commentary on
Book I was written he was evidently uncertain whether he
would be able to continue it, for at the end he says, ' For my
part, if I should be able to discuss the other Books in the
same way, I should give thanks to the gods ; but, if other
cares should draw me away, I beg those who are attracted by
this subject to complete the exposition of the other Books as
well, following the same method and addressing themselves
throughout to the deeper and more sharply defined questions
involved '} Wachsmuth, finding a Vatican manuscript contain-
ing a collection of scholia on Books I, II, V, VI, X, headed Eis ra
EvKXeiSou (TToi)(€ia TTpoXanfiavojieva €K toou UpoKXov (nropdSrju
Kal kolt €7TLTo/xriu, and seeing that the scholia on Book I were
extracts from the extant commentary of Proclus, concluded
that those on the other Books were also from Proclus; but
the 7r/oo- in TTpoXa\x$avbii£va rather suggests that only the
scholia to Book I are from Proclus. Heiberg found and
published in 1903 a scholium to X. 9, in which Proclus is
expressly quoted as the authority, but he does not regard
this circumstance as conclusive. On the other hand, Heiberg
has noted two facts which go against the view that Proclus
wrote on the later Books: (1) the scholiast's copy of
Proclus was not much better than our manuscripts ; in
particular, it had the same lacunae in the notes to I. 36,
37, and I. 41-3; this makes it improbable that the scholiast
had further commentaries of Proclus which have vanished
for us ; (2) there is no trace in the scholia of the notes
which Proclus promised in the passages already referred to.
All, therefore, that we can say is that, while the Wachsmuth
scholia may be extracts from Proclus, it is on the whole
improbable.
Hypotyposis of Astronomical Hypotheses.
Another extant work of Proclus which should be referred
to is his Hypotyposis of Astronomical Hypotheses, a sort of
readable and easy introduction to the astronomical system
of Hipparchus and Ptolemy. It has been well edited by
Manitius (Teubner, 1909). Three things may be noted as
1 Proclus on Eucl. I, p. 432. 9-15.
536 COMMENTATORS AND BYZANTINES
regards this work. It contains 1 a description of the method
of measuring the sun's apparent diameter by means of
Heron's water-clock, which, by comparison with the corre-
sponding description in Theon's commentary to the Syntaxis
of Ptolemy, is seen to have a common source with it. That
source is Pappus, and, inasmuch as Proclus has a figure (repro-
duced by Manitius in his text from one set of manuscripts)
corresponding to the description, while the text of Theon has
no figure, it is clear that Proclus drew directly on Pappus,
who doubtless gave, in his account of the procedure, a figure
taken from Heron's own work on water-clocks. A simple
proof of the equivalence of the epicycle and eccentric hypo-
theses is quoted by Proclus from one Hilarius of Antioch. 2
An interesting passage is that in chap. 4 (p. 130, 18) where
Sosigenes the Peripatetic is said to have recorded in his work
1 on reacting spheres ' that an annular eclipse of the sun is
sometimes observed at times of perigee ; this is, so far as
I know, the only allusion in ancient times to annular eclipses,
and Proclus himself questions the correctness of Sosigenes's
statement.
Commentary on the Republic.
The commentary of Proclus on the Republic contains some
passages of great interest to the historian of mathematics.
The most important is that 3 in which Proclus indicates that
Props. 9, 10 of Euclid, Book II, are Pythagorean proposi-
tions invented for the purpose of proving geometrically the
fundamental property of the series of ' side-' and ' diameter-'
numbers, giving successive approximations to the value of
\/2 (see vol. i, p. 93). The explanation 4 of the passage in
Plato about the Geometrical Number is defective and dis-
appointing, but it contains an interesting reference to one
Paterius, of date presumably intermediate between Nestorius
and Proclus. Paterius is said to have made a calculation, in
units and submultiples, of the lengths of different segments of
1 Proclus, Hypotyposis, c. 4, pp. 120-22.
2 lb., c. 3, pp. 76, 17 sq.
3 Prodi Diadochi in Platonis Rempublicam Commentarii, ed. Kroil,
vol. ii, p. 27.
4 lb., vol. ii, pp. 36-42.
PROCLUS. MARINUS 537
straight lines in a figure formed by taking a triangle with
sides 3, 4, 5 as ABG, then drawing
BD from the right angle B perpen-
dicular to AG, and lastly drawing-
perpendiculars BE, BF to AB, BC.
A diagram in the text with the
lengths of the segments shown along-
side them in the usual numerical
notation shows that Paterius obtained from the data AB = 3,
BG = 4, CA = 5 the following :
£Z) = |S/ lf '=2^ [=2|]
42>=«S«V = lHA [=lf]
w = /3 s *y = 2f j T i 5 [=2|f]
^ = .^^=11^* [=i«]
BE=aSy' »? v' = 1H A A [ = Iff]
This is an example of the Egyptian method of stating frac-
tions preceding by some three or four centuries the exposition
of the same method in the papyrus of Akhmim.
Marinus of Neapolis, the pupil and biographer of Proclus,
wrote a commentary or rather introduction to the Data of
Euclid. 1 It is mainly taken up with a discussion of the
question ri to deSo/ievov, what is meant by given 1 There
were apparently many different definitions of the term given
by earlier and later authorities. Of those who tried to define
it in the simplest way by means of a single differentia, three
are mentioned by name. Apollonius in his work on vevaei?
and his ' general treatise ' (presumably that on elementary
geometry) described the given as assigned or fixed (reTay-
\xkvov), Diodorus called it kno%vn (yvdopifiov); others regarded
it as rational {p-qrov) and Ptolemy is classed with these, rather
oddly, because ' he called those things given the measure of
which is given either exactly or approximately'. Others
1 See Heiberg and Menge's Euclid, vol. vi, pp. 234-56.
538 COMMENTATORS AND BYZANTINES
combined two of these ideas and called it assigned or fixed
and procurable or capable of being found (nopifioit); others
' fixed and known ', and a third class ' known and procurable '.
These various views are then discussed at length.
Domninus of Larissa, a pupil of Syrianus at the same time
as -Proclus, wrote a Manual of Introductory Arithmetic eyx^-
piSiov dpiOfj.r]TiKrjs elaayooyfjs, which was edited by Boissonade 1
and is the subject of two articles by Tannery, 2 who also left
a translation of it, with prolegomena, which has since been
published. 3 It is a sketch of the elements of the theory of
numbers, very concise and well arranged, and is interesting
because it indicates a serious attempt at a reaction against the
Introductio arithmetica of Nicomachus and a return to the
doctrine of Euclid. Besides Euclid, Nicomachus and Theon
of Smyrna, Domninus seems to have used another source,
now lost, which was also drawn upon by Iamblichus. At the
end of this work Domninus foreshadows a more complete
treatise oh the theory of numbers under the title Elements of
Arithmetic (dpiO/jirjTiKr} a-Toi^e'iadcns;), but whether this was
ever written or not we do not know. Another tract
attributed to Domninus ttco? eari \6yov e/c Xoyov afeXelv
(how a ratio can be taken out of a ratio) has been published
with a translation by Ruelle 4 ; if it is not by Domninus, it
probably belongs to the same period.
A most honourable place in our history must be reserved
for Simplicius, who has been rightly called ' the excellent
Simplicius, the Aristotle-commentator, to whom the world can
never be grateful enough for the preservation of the frag-
ments of Parmenides, Empedocles, Anaxagoras, Melissus,
Theophrastus and others' (v. Wilamowitz-Mollendorff). He
lived in the first half of the sixth century and was a pupil,
first of Ammonius of Alexandria, and then of Damascius,
the last head of the Platonic school at Athens. When in the
year 529 the Emperor Justinian, in his zeal to eradicate
paganism, issued an edict forbidding the teaching of philo-
1 Anecdota Graeca, vol. iv, pp. 413-29.
2 Me'moires scientifiques, vol. ii, nos. 35, 40.
3 Revue des etudes grecques, 1906, pp. 359-82; Memoires scientifiques^
vol. iii, pp. 256-81.
4 Revue de Philologie, 1883, p. 83 sq.
DOMNINUS. SIMPLICIUS 539
sophy at Athens, the last members of the school, including
Damascius and Simplicius, migrated to Persia, but returned
about 533 to Athens, where Simplicius continued to teach for
some time though the school remained closed.
Extracts from Eudemus.
To Simplicius we owe two long extracts of capital impor-
tance for the history of mathematics and astronomy. The
first is his account, based upon and to a large extent quoted
textually from Eudemus's History of Geometry, of the attempt
by Antiphon to square the circle and of the quadratures of
lunes by Hippocrates of Chios. It is contained in Simplicius's
commentary on Aristotle's Physics} and has been the subject
of a considerable literature extending from 1870, the date
when Bretschneider first called attention to it, to the latest
critical edition with translation and notes by Rudio (Teubner,
1907). It has already been discussed (vol. i, pp. 183-99).
The second, and not less important, of the two passages is
that containing the elaborate and detailed account of the
system of concentric spheres, as first invented by Eudoxus for
explaining the apparent motion of the sun, moon, and planets,
and of the modifications made by Callippus and Aristotle. It
is contained in the commentary on Aristotle's Be caelo * ;
Simplicius quotes largely from Sosigenes the Peripatetic
(second century a.d.) 5 observing that he in his turn drew
from Eudemus, who dealt with the subject in the second
book of his History of Astronomy. It is this passage of
Simplicius which, along with a passage in Aristotle's Meta-
physics? enabled Schiaparelli to reconstruct Eudoxus' s system
(see vol. i, pp. 329-34). Nor must it be forgotten that it is in
Simplicius's commentary on the Physics^ that the extract
from Geminus's summary of the Meteorologica of Posidonius
occurs which was used by Schiaparelli to support his view
that it was Heraclides of Pontus, not Aristarchus of Samos,
who first propounded the heliocentric hypothesis.
Simplicius also wrote a commentary on Euclid's Elements,
Book I, from which an-Nairizi, the Arabian commentator,
1 Simpl. in Phtjs., pp. 54-69, ed. Diels.
2 Simpl. on Arist. De caelo, p. 488. 18-24 and pp. 493-506, ed. Heiberg.
3 Metaph. A. 8, 1073 b 17-1074 a 14.
4 Simpl. in Phys., pp. 291-2, ed. Diels.
540 COMMENTATORS AND BYZANTINES
made valuable extracts, including the account of the attempt of
'Aganis' to prove the parallel-postulate (see pp. 228 30 above).
Contemporary with Simplicius, or somewhat earlier, was
Eutocius, the commentator on Archimedes and Apollonius.
As he dedicated the commentary on Book I On the Sphere
and Cylinder to Ammonius (a pupil of Proclus and teacher
of Simplicius), who can hardly have been alive after a.d. 510,
Eutocius was probably born about A.D. 480. His date used
to be put some fifty years later because, at the end of the com-
mentaries on Book II On the Sphere and Cylinder and on
the Measurement of a Circle, there is a note to the effect that
' the edition was revised by Isidorus of Miletus, the mechanical
engineer, our teacher '. But, in view of the relation to Ammo-
nius, it is impossible that Eutocius can have been a pupil of
Isidorus, who was younger than Anthemius of Tralles, the
architect of Saint Sophia at Constantinople in 532, whose
work was continued by Isidorus after Anthemius's death
about a.d. 534. Moreover, it was to Anthemius that Eutocius
dedicated, separately, the commentaries on the first four
Books of Apollonius's Conies, addressing Anthemius as ' my
dear friend '. Hence we conclude that Eutocius was an elder
contemporary of Anthemius, and that the reference to Isidorus
is by an editor of Eutocius's commentaries who was a pupil of
Isidorus. For a like reason, the reference in the commentary
on Book II On the Sphere and Cylinder 1 to a SiafirJTrjs
invented by Isidorus ' our teacher ' for drawing a parabola
must be considered to be an interpolation by the same editor.
Eutocius's commentaries on Archimedes apparently ex-
tended only to the three works, On the Sphere and Cylinder,
Measurement of a Circle and Plane Equilibriums, and those
on the Conies of Apollonius to the first four Books only.
We are indebted to these commentaries for many valuable
historical notes. Those deserving special mention here are
(1) the account of the solutions of the problem of the duplica-
tion of the cube, or the finding of two mean proportionals,
by ' Plato ', Heron, Philon, Apollonius, Diodes, Pappus,
Sporus, Menaechmus, Archy tas, Eratosthenes, Nicomedes, (2)
the fragment discovered by Eutocius himself containing the
1 Archimedes, ed. Heiberg, vol. iii, p. 84. 8-11.
EUTOCIUS. ANTHEMIUS 541
missing solution, promised by Archimedes in On the Sphere
and Cylinder, II. 4, of the auxiliary problem amounting
to the solution by means of conies of the cubic equation
(a — x)x 2 = be 2 , (3) the solutions (a) by Diocles of the original
problem of II. 4 without bringing in the cubic, (b) by Diony-
sodorus of the auxiliary cubic equation.
Anthemius of Tralles, the architect, mentioned above, was
himself an able mathematician, as is seen from a fragment of
a work of his, On Burning-mirrors. This is a document of
considerable importance for the history of conic sections.
Originally edited by L. Dupuy in 1777, it was reprinted in
Westermann's IIapaSo£oy pd(f>oi (Scriptores rerum mirabiliwm
Graeci), 1839, pp. 14 9-58. The first and third portions of
the fragment are those which interest us. 1 The first gives
a solution of the problem, To contrive that a ray of the sun
(admitted through a small hole or window) shall fall in a
given spot, without moving away at any hour and season.
This is contrived by constructing an elliptical mirror one focus
of which is at the point where the ray of the sun is admitted
while the other is at the point to which the ray is required
to be reflected at all times. Let B be the hole, A the point
to which reflection must always take place, BA being in the
meridian and parallel to the horizon. Let BO be at right
angles to BA, so that OB is an equinoctial ray ; and let BD be
the ray at the summer solstice, BE a winter ray.
Take F at a convenient distance on BE and measure FQ
equal to FA. Draw HFG through F bisecting the angle
AFQ, and let BG be the straight line bisecting the angle EBO
between the winter and the equinoctial rays. Then clearly j
since FG bisects the angle QFA, if we have a plane mirror in
the position HFG, the ray BFE entering at B will be reflected
to J..
To get the equinoctial ray similarly reflected to A, join GA,
and with G as centre and GA as radius draw a circle meeting
BO in K. Bisect the angle KG A by the straight line GLM
meeting BK in L and terminated at 31, a point on the bisector
of the angle CBD. Then LM bisects the angle KLA also, and
KL = LA, and KM = MA. If then GLM is a plane mirror,
the ray BL will be reflected to A.
1 See Bibliotheca mathematica, vii 3 , 1907, pp. 225-33.
542
COMMENTATORS AND BYZANTINES
By taking the point AT on BD such that MN = MA, and
bisecting the angle NMA by the straight line MOP meeting
BD in 0, we find that, if MOP is a plane mirror, the ray BO
is reflected to A.
Similarly, by continually bisecting angles and making more
mirrors, we can get any number of other points of impact. Mak-
ing the mirrors so short as to form a continuous curve, we get
the curve containing all points such that the sum of the distances
of each of them from A and B is constant and equal to BQ, BK,
or BN. ' If then ', says Anthemius, ' we stretch a string passed
round the points A, B, and through the first point taken on the
rays which are to be reflected, the said curve will be described,
which is part of the so-called " ellipse ", with reference to
which (i.e. by the revolution of which round BA) the surface
of impact of the saichmirror has to be constructed*'
We have here apparently the first mention of the construc-
tion of an ellipse by means of a string stretched tight round
the foci. Anthemius's construction depends upon two pro-
positions proved by Apollonius (1) that the sum of the focal
distances of any point on the ellipse is constant, (2) that the
focal distances of any point make equal angles with the
tangent at that point, and also (3) upon a proposition not
found in Apollonius, namely that the straight line joining
ANTHEMIUS 543
the focus to the intersection of two tangents bisects the angle
between the straight lines joining the focus to the two points
of contact respectively.
In the third portion of the fragment Anthemius proves that
parallel rays can be reflected to one single point from a para-
bolic mirror of which the point is the focus. The directrix is
used in the construction, which follows, mutatis mutandis, the
same course as the above construction in the case of the ellipse.
As to the supposition of Heiberg that Anthemius may also
be the author of the Fragmentum mathematicum Bobiense, see
above (p. 203).
. The Papyrus of Akhmvm.
Next in chronological order must apparently be placed
the Papyrus of Akhmlm, a manual of calculation written
in Greek, which was found in the metropolis of Akhmim,
the ancient Panopolis, and is now in the Musee du
Gizeh. It was edited by J. Baillet l in 1892. Accord-
ing to the editor, it was written between the sixth and
ninth centuries by a Christian. It is interesting because
it preserves the Egyptian method of reckoning, with proper
fractions written as the sum of primary fractions or sub-
multiples, a method which survived alongside the Greek and
was employed, and even exclusively taught, in the East. The
advantage of this papyrus, as compared with Ahmes's, is that
we can gather the formulae used for the decomposition of
ordinary proper fractions into sums of submultiples. The
formulae for decomposing a proper fraction into the sum of
two submultiples may be shown thus :
, . a 1 1
0) t=-t-t. +
be b + c 7 b + c
c . b .
a a
_ 2 11 3 1 1 18 11
Examples — = » — = ? = •
F 11 666 110 7077 323 34 38
, , a 1 1
be b + mc , b + mc 1
c. b.
a am
1 Memoires publies par les membres de la Mission archeologique frangaise
au Caire, vol. ix, part 1, pp. 1-89.
544 COMMENTATORS AND BYZANTINES
E J^ 1 1 1^ 3
X * 176 ~" /16 + 3 . 11\ + /16 + 3. 11\ 1~ = 77 + 112'
U (—7^-) 16 (-^)3
3 1 1 11
and again — = H =
6 112 /16 + 2.7\ /16 + 2.7\1 70 80
7 (-3—) 16 (— r~ )i
a 1 1
/3) == 1 .
' cdf cd + df f cd + df
a a
Example.
28 28 1 111
+
1320 10.12.11" 120 + 132 120+132 90 99
' 28~ ' 28
The object is, of course, to choose the factors of the denomi-
nator, and the multiplier m in (2), in such a way as to make
the two denominators on the right-hand side integral.
When the fraction has to be decomposed into a sum of three
or more submultiples, we take out an obvious submultiple
first, then if necessary a second, until one of the formulae
will separate what remains into two submultiples. Or we
take out a part which is not a submultiple but which can be
divided into two submultiples by one of the formulae.
For example, to decompose -^j^. The factors of 61 6 are 8.7 7
or 7 . 88. lake out gg, and ^ T e = gg 6 T6 - = 8 8 7 7 = 88 77 11 5
and T 2 T = eV A by formula (1), so that ^ = -£■? T V is A •
Take ^V The factors of 6460 are 85.76 or 95.68. Take
out q- 1 ^, and 6 2 4 3 6 9 o = st ihtwo • Again take out 93-, and we have
ws 9*5 eif or is 9V is • r ^ ie actual problem here is to find
3^3 rd of HJ^to eV, which latter expression reduces to
20 • ^39.
The sort of problems solved in the book are (1) the division
of a number into parts in the proportion of certain given
numbers, (2) the solution of simple equations such as this:
From a certain treasure we take away j^th, then from the
remainder y 7 th of that remainder, and we find 150 units left;
what was the treasure? \<x x — t(x x\ — ,,, > =.R.
THE PAPYRUS OF AKHMIM. PSELLUS 545
(3) subtractions such as: From § subtract iVttiVAhis
1111111. Ill 1 1 1 An«wpr JL JL
40 44 5 55 60 66 7 7 7 88 9 95 100 110* auBVVC1 ) 10 50'
The book ends with long tables of results obtained (1) by
multiplying successive numbers, tens, hundreds and thousands
up to 10,000 by §, -§, i, J, |, &c, up to ^, (2) by multiplying
all the successive numbers 1, 2, 3 ... n by -> where n is succes-
sively 11, 12, ... and 20; the results are all arranged as the
sums of integers and submultiples.
The Geodaesia of a Byzantine author formerly called, with-
out any authority, ' Heron the Younger ' was translated into
Latin by Barocius in 1572, and the Greek text was published
with a French translation by Vincent. 1 The place of the
author's observations was the hippodrome at Constantinople,
and the date apparently about 938. The treatise was modelled
on Heron of Alexandria, especially the Dioptra, while some
measurements of areas and volumes are taken from the
Metrica.
Michael Psellus lived in the latter part of the eleventh
century, since his latest work bears the date 1092. Though
he was called ' first of philosophers ', it cannot be said that
what survives of his mathematics suits this title. Xylander
edited in 1556 the Greek text, with a Latin translation, of
a book purporting to be by Psellus on the four mathematical
sciences, arithmetic, music, geometry and astronomy, but it is
evident that it cannot be entirely Psellus's own work, since
the astronomical portion is dated 1008. The arithmetic con-
tains no more than the names and classification of numbers
and ratios. The geometry has the extraordinary remark that,
while opinions differed as to how to find the area of a circle,
the method which found most favour was to take the area as
the geometric mean between the inscribed and circumscribed
squares; this gives tt = </8 =2-8284271 ! The only thing of
Psellus which has any value for us is the letter published by
Tannery in his edition of Diophantus. 2 In this letter Psellus
says that both Diophantus and Anatolius (Bishop of Laodicea
about A. D. 280) wrote on the Egyptian method of reckoning,
1 Notices et extraits, xix, pt. 2, Paris, 1858.
2 Diophantus, vol. ii, pp. 37-42.
1523.2 n n
546 COMMENTATORS AND BYZANTINES
and that Anatolius's account, which was different and more
succinct, was dedicated to Diophantus (this enables us to
determine Diophantus's date approximately). He also notes
the difference between the Diophantine and Egyptian names
for the successive powers of dpiOfjios : the next power after
the fourth (Svvafj.o8vvaiJ.is = x^), i.e. x 5 , the Egyptians called
' the first undescribed ' (aXoyos irpcoros) or the ' fifth number ' ;
the sixth, x 6 , they apparently (like Diophantus) called the
cube -cube ; but with them the seventh, x 1 , was the ' second
undescribed ' or the ' seventh number ', the eighth (x*) was the
' quadruple square ' (reTpairXfj Swa/its), the ninth (x d ) the
'extended cube' (kv&os k^tXiKros). Tannery conjectures that
all these remarks were taken direct from an old commentary
on Diophantus now lost, probably Hypatia's.
Georgius Pachymeres (1242-1310) was the author of a
work on the Quadrivium (SvvTayjia rcov Tea-crdpcov fiaOrj/jLarcou
or TerpafiifiXov). The arithmetical portion contains, besides
excerpts from Nicomachus and Euclid, a paraphrase of Dio-
phantus, Book I, which Tannery published in his edition of
Diophantus l ; the musical section with part of the preface was
published by Vincent, 2 and some fragments from Book IV by
Martin in his edition of the Astronomy of Theon of Smyrna.
Maximus Planudes, a monk from Nicomedia, was the
envoy of the Emperor Andronicus II at Venice in the year
1297, and lived probably from about 1260 to 1310. He
wrote scholia on the first two Books of Diophantus, which
are extant and are included in Tannery's edition of Dio-
phantus. 3 They contain nothing of particular interest except
a number of conspectuses of the working-out of problems of
Diophantus written in Diophantus's own notation but with
steps in separate lines, and with abbreviations on the left of
words indicating the operations (e.g. €k0. = e/c^eo-ij, rerp. =
TeTpayodvio-fios, <rvv0. — o-vvOeais, &c.); the result is to make
the work almost as easy to follow as it is in our notation.
Another work of Planudes is called Wr)(po(popia KaT 'IvSovs,
or Arithmetic after the Indian method, and was edited as Das
1 Diophantus, vol. ii, pp. 78-122.
2 Notices et extraits, xvii, 1858, pp. 362-533.
3 Diophantus, vol. ii, pp. 125-255.
PSELLUS. PACHYMERES. PLANUDES 547
Rechenbuch des Maximus Planudes in Greek by Gerharclt
(Halle, 1805) and in a German translation by H. Waeschke
(Halle, 1878). There was, however, an earlier book under the
similar title 'Apyrj Trjs fjieydXr)? kccl 'IvSiktjs y\rr]^>L(j)opLas (sic),
written in 1252, which is extant in the Paris MS. Suppl. Gr.
387 ; and Planudes seems to have raided this work. He
begins with an account of the symbols which, he says, were
' invented by certain distinguished astronomers for the most
convenient and accurate expression of numbers. There are
nine of these symbols (our 1, 2, 3, 4, 5, 6, 7, 8, 9), to which is
added another called Tzifra (cypher), written and denoting
zero. The nine signs as well as this one are Indian.'
But this is, of course, not the first occurrence of the Indian
numerals; they were known, except the zero, to Gerbert
(Pope Sylvester II) in the tenth century, and were used by
Leonardo of Pisa in his Liber abaci (written in 1202 and
rewritten in 1228). Planudes used the Persian form of the
numerals, differing in this from the writer of the treatise of
1252 referred to, who used the form then current in Italy.
It scarcely belongs to Greek mathematics to give an account
of Planudes's methods of subtraction, multiplication, &c.
Extraction of the square root.
As regards the extraction of the square root, he claims to
have invented a method different from the Indian method
and from that of Theon. It does not appear, however, that
there was anything new about it. Let us try to see in what
the supposed new method consisted.
Planudes describes fully the method of extracting the
square root of a number with several digits, a method which
is essentially the same as ours. This appears to be what he
refers to later on as ' the Indian method '. Then he tells us
how to find a first approximation to the root when the number
is not a complete square.
' Take the square root of the next lower actual square
number, and double it : then, from the number the square root
of which is required, subtract the next lower square number
so found, and to the remainder (as numerator) give as de-
nominator the double of the square root already found.'
N n 2
548 COMMENTATORS AND BYZANTINES
The example given is -/(18). Since 4 2 = 16 is the next
2
lower square, the approximate square root is 4 + - — or 4J.
The formula used is, therefore, \/(a 2 + b) = a + — approxi-
mately. (An example in larger numbers is
\/(1690196789) = 41112 + §111* approximately.)
Planudes multiplies 4^ by itself and obtains 18^, which
shows that the value 4 J is not accurate. He adds that he will
explain later a method which is more exact and nearer the
truth, a method ' which I claim as a discovery made by me
with the help of God '. Then, coming to the method which he
claims to have discovered, Planudes applies it to V§. The
object is to develop this in units and sexagesimal fractions.
Planudes begins by multiplying the 6 by 3600, making 21600
second-sixtieths, and finds the square root of 21600 to lie
between 146 and 147. Writing the 146' as 2 26', he proceeds
to find the rest of the approximate square root (2 26' 58" 9'")
by the same procedure as that used by Theon in extracting
the square root of 4500 and 2 28' respectively. The differ-
ence is that in neither of the latter cases does Theon multiply
by 3600 so as to reduce the units to second-sixtieths, but he
begins by taking the approximate square root of 2, viz. 1, just
as he does that of 4500 (viz. 67). It is, then, the multiplication
by 3600, or the reduction to second-sixtieths to start with, that
constitutes the difference from Theon's method, and this must
therefore be what Planudes takes credit for as a new dis-
covery. In such a case as V(2 28') or >/3, Theon's method
has the inconvenience that the number of minutes in the
second term (34' in the one case and 43' in the other) cannot
be found without some trouble, a difficulty which is avoided
by Planudes's expedient. Therefore the method of Planudes
had its advantage in such a case. But the discovery was not
new. For it will be remembered that Ptolemy (and doubtless
Hipparchus before him) expressed the chord in a circle sub-
tending an angle of 120° at the centre (in terms of 120th parts
of the diameter) as 103 p 55' 23", which indicates that the first
step in calculating Vs was to multiply it by 3600, making
10800, the nearest square below which is 103 2 (— 10609). In
PLANUDES. MOSCHOPOULOS 549
the scholia to Eucl., Book X, the same method is applied.
Examples have been given above (vol. i, p. 63). The supposed
new method was therefore not only already known to the
scholiast, but goes back, in all probability, to Hipparchus.
Two problems.
Two problems given at the end of the Manual of Planudes
are worth mention. The first is stated thus : ' A certain man
finding himself at the point of death had his desk or safe
brought to him and divided his money among his sons with
the following words, " I wish to divide my money equally
between my sons : the first shall have one piece and ^th of the
rest, the second 2 and ^th of the remainder, the third 3 and
\ th of the remainder." At this point the father died without
getting to the end either of his money or the enumeration of
his sons. I wish to know how many sons he had and how
much money.' The solution is given as (n — l) 2 for the number
of coins to be divided and (n — 1 ) for the number of his sons ;
or rather this is how it might be stated, for Planudes takes
n = 7 arbitrarily. Comparing the shares of the first two we
must clearly have
1 1 /Y> 1
l+-(®-l) = 2 + -{ X -(l+- + 2)},
which gives x = (n — l) 2 ; therefore each of (n — 1) sons received
(n-l).
The other problem is one which we have already met with,
that of finding two rectangles of equal perimeter such that
the area of one of them is a given multiple of the area of
the other. If n is the given multiple, the rectangles are
(n 2 — 1, n 3 — n 2 ) and (n— 1, n 3 — n) respectively. Planudes
states the solution correctly, but how he obtained it is not clear.
We find also in the Manual of Planudes the ' proof by nine '
(i.e. by casting out nines), with a statement that it was dis-
covered by the Indians and transmitted to us through the
Arabs.
Manuel Moschopoulos, a pupil and friend of Maximus
Planudes, lived apparently under the Emperor Andronicus II
(1282-1328) and perhaps under his predecessor Michael VIII
(1261-82) also. A man of wide learning, he wrote (at the
550 COMMENTATORS AND BYZANTINES
instance of Nicolas Rhabdas, presently to be mentioned) a
treatise on magic squares ; he showed, that is, how the num-
bers 1 , 2, 3 . . . n 2 could be placed in the n 2 compartments of
a square, divided like a chess-board into n 2 small squares, in such
a way that the sum of the numbers in each horizontal and
each vertical row of compartments, as well as in the rows
forming the diagonals, is always the same, namely \n (n 2 + 1).
Moschopoulos gives rules of procedure for the cases in which
n = 2 m + 1 and n = 4 m respectively, and these only, in the
treatise as we have it ; he promises to give the case where
n = 4m+2 also, but does not seem to have done so, as the
two manuscripts used by Tannery have after the first two cases
the words reAo? rod avrov. The treatise was translated by
De la Hire, 1 edited by S. Gunther, 2 and finally edited in an
improved text with translation by Tannery. 3
The work of Moschopoulos was dedicated to Nicolas Arta-
vasdus, called Rhabdas, a person of some importance in the
history of Greek arithmetic. He edited, with some additions
of his own, the Manual of Planudes; this edition exists in
the Paris MS. 2428. But he is also the author of two letters
which have been edited by Tannery in the Greek text with
French translation. 4 The date of Rhabdas is roughly fixed
by means of a calculation of the date of Easter ' in the current
year ' contained in one of the letters, which shows that its
date was 1341. It is remarkable that each of the two letters
has a preface which (except for the words tt]v 8rj\a>cru/ ra>u kv
roh dpiOfioT? £r)Trjfj,dTcov and the name or title of the person
to whom it is addressed) copies word for word the first thir-
teen lines of the preface to Diophantus's Arithmetica, a piece
of plagiarism which, if it does not say much for the literary
resource of Rhabdas, may indicate that he had studied Dio-
phantus. The first of the two letters has the heading ' A con-
cise and most clear e